UNIVERSITY  OF  CALIFORNIA 

AT   LOS  ANGELES 


PHILLIPS-LOOMIS  MATHEMATICAL    SERIES 

ELEMENTS   OF   TRIGONOMETKY 
WITH    TABLES 


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PHILLIPS-LOOMIS  MATHEMATICAL   SERIES 


ELEMENTS    OF    TRIGONOMETRY 


PLANE    AND    SPHERICAL 


BY 


ANDREW    W.   PHILLIPS,   Ph.D. 

AND 

WENDELL   M.  STRONG,  Ph.D. 

YALE   UNIVERSITY 


45071 


NEW   YORK   AND    LONDON 

HARPER    &     BROTHERS     PUBLISHERS 

1899 


THE  PHILLIPS-LOOMIS  MATHEMATICAL  SERIES. 


ELEMENTS  OF  TRIGONOMETRY,  Plane  and  Spherical.  By 
Andkew  W.  Phillips,  Ph.D.,  and  Wendell  M.  Strong,  Ph.D.,  Yale 
University.    Crown  8vo,  Half  Leather. 

ELEMENTS  OF  GEOMETRY.  By  Andrew  W.  Phillips,  Ph.D., 
and  Irving  Fisher,  Ph.D.,  Professors  in  Yale  University.  Crown 
8vo,  Half  Leather,  $1  75.    [By  mail,  $1  92.] 

ABRIDGED  GEOMETRY.  By  Andrew  W.  Phillips,  Ph.D.,  and 
Irving  Fisher,  Ph.D.  Crown  8vo,  Half  Leather,  $1  25.  [By 
mail,  $1  40.] 

PLANE  GEOMETRY.  By  Andrew  W.  Phillips,  Ph.D.,  and  Irvi.ng 
Fisher,  Ph.D.    Crown  Bvo,  Cloth,  80  cents.    [By  mail,  90  cents.] 

LOGARITHMIC  AND  TRIGONOMETRIC  TABLES.  Five-Place 
and  Four- Place.  By  Andrew  W.  Phillips,  Ph.D.,  and  Wendell 
M.  Strong,  Ph.D.,  Yale  University.    Crown  8vo. 

LOGARITHMS  OF  NUMBERS.  Five-Figure  Table  to  Accompany 
the  "Elements  of  Geometry,"  by  Andrew  W.  Phillips,  Ph.D.,  and 
Irving  Fisher,  Ph.D.,  Professors  in  Yale  University.  Crown  8vo, 
Cloth,  30  cents.    [By  mail,  35  cents.] 


NEW    YORK    AND   LONDON  : 
HARPER    &    BROTHERS,   PUBLISHERS. 


Copyright,  1898,  by  Harper  &  Brothers. 

y4/i  rights  reserved. 


Mathematical  (^  A 
Sciences 
Library     O  o  \ 


PREFACE 


P^-^ 


In  this  work  the  trigonometric  functions  are  defined  as 
I  ratios,  but  their  representation  by  lines  is  also  introduced  at 
1  the  beginning,  because  certain  parts  of  the  subject  can  be 
^treated  more  simply  by  the  line  method,  or  by  a  combination 
\of  the  two  methods,  than  by  the  ratio  method  alone. 

Attention  is  called  to  the  following  features  of  the  book : 
The  simplicity  and  directness  of  the  treatment  of  both 
the  Plane  and  Spherical  Trigonometry. 
J      The  emphasis  given  to  the  formulas  essential  to  the  solu- 
.  tion  of  triangles.  "  ,  • 

V      The  large  number  of  exercises. 
^'       The  graphical  representation  of  the  trigonometric,  inverse 
trigonometric,  and  hyperbolic  functions. 

The  use  of  photo-engravings  of  models  in  the  Spherical 
Trigonometry. 

The  recognition  of  the  rigorous  ideas  of  modern  math- 

Jematics  in   dealing  with   the   fundamental  series  of  trigo- 
nometry. 
The  natural  treatment  of  the  complex  number  and  the 
hyperbolic  functions. 

The  graphical  solution  of  spherical  triangles. 
Our  grateful  acknowledgments  are  due  to  our  colleague, 
Professor  James  Pierpont,  for  valuable  suggestions  regard- 
ing the  construction  of  Chapter  VI. 

We  are  also  indebted  to  Dr.  George  T.  Sellew  for  making 
the  collection  of  miscellaneous  exercises. 

Andrew  W.  Phillips, 
Wendell  M.  Strong. 
Yale  University,  December,  iSgS. 


TABLE    OF   CONTENTS 


PLANE   TRIGONOMETRY 
CHAPTER  I 

THE  TRIGONOMETRIC   FUNCTIONS 

PAGE 

Angles I 

Definitions  of  the  Trigonometric  Functions 4 

Signs  of  the  Trigonometric  Functions 8 

Relations  of  the  Functions 10 

Functions  of  an  Acute  Angle  of  a  Right  Triangle 13 

Functions  of  Complementary  Angles 14 

Functions  of  0°,  90°,  180°,  270°  360° 15 

Functions  of  the  Supplement  of  an  Angle 16 

Functions  of  45°,  30°,  60° 17 

Functionsof  (  — ^),  (1800— ;t-),  (i8o04-;r),  (360°— ;tr) 18 

Functions  of  (90°—/),  (90°+/),  {270°— y),  {27o°-\-y) 20 

CHAPTER   II 

THE  RIGHT  TRIANGLE 

Solution  of  Right  Triangles 22 

Solution  of  Oblique  Triangles  by  the  Aid  of  Right  Triangles    .    .  28 

CHAPTER   III 

TRIGONOMETRIC   ANALYSIS 

Proof  of  Fundamental  Formulas  (11)- (14) 32 

Tangent  of  the  Sum  and  Difference  of  Two  Angles 36 

Functions  of  Twice  an  Angle 36 

Functions  of  Half  an  Angle 36 

Formulas  for  the  Sums  and  Differences  of  Functions 37 

The  Inverse  Trigonometric  Functions 39 


vi  TABLE  OF  CONTENTS 

CHAPTER   IV 

THE  OBLIQUE  TRIANGLE 

PAGE 

Derivation  of  Formulas 41 

Formulas  for  the  Area  of  a  Triangle 44 

The  Ambiguous  Case 45 

The  Solution  of  a  Triangle  : 

(I.)  Given  a  Side  and  Two  Angles 46 

(2.)  Given  Two  Sides  and  the  Angle  Opposite  One  of  Them     .  46 

(3.)  Given  Two  Sides  and  the  Included  Angle 48 

(4.)  Given  the  Three  Sides 49 

Exercises    .    .    * 50 

CHAPTER  V 

CIRCULAR  MEASURE — GRAPHICAL   REPRESENTATION 

Circular  Measure 55 

Periodicity  of  the  Trigonometric  Functions 57 

Graphical  Representation 58 

CHAPTER  VI 

COMPUTATION  OF  LOGARITHMS  AND  OF  THE  TRIGONOMETRIC  FUNC- 
TIONS—DE   MOIVRE'S  THEOREM— HYPERBOLIC   FUNCTIONS 

Fundamental  Series 63 

Computation  of  Logarithms 64 

Computation  of  Trigonometric  Functions 68 

De  Moivre's  Theorem 70 

The  Roots  of  Unity ■ 72 

The  Hyperbolic  Functions 73 

CHAPTER  VII 

MISCELLANEOUS  EXERCISES 

Relations  of  Functions 78 

Right  Triangles 80 

Isosceles  Triangles  and  Regular  Polygons 83 

Trigonometric  Identities  and  Equations 84 

Oblique  Triangles 88 


TABLE  OF  CONTENTS  vii 

SPHERICAL   TRIGONOMETRY 
CHAPTER  VIII 

RIGHT   AND   QUADRANTAL  TRIANGLES 

PAGE 

Derivation  of  Formulas  for  Right  Triangles 93 

Napier's  Rules 94 

Ambiguous  Case 97 

Quadrantal  Triangles 98 

CHAPTER    IX 

OBLIQUE-ANGLED   TRIANGLES 

Derivation  of  Formulas 100 

Formulas  for  Logarithmic  Computation loi 

The  Six  Cases  and  Examples 104 

Ambiguous  Cases 106 

Area  of  the  Spherical  Triangle 108 

CHAPTER   X 

APPLICATIONS   TO   THE  CELESTIAL  AND  TERRESTRIAL   SPHERES 

Astronomical  Problems no 

Geographical  Problems 113 

CHAPTER  XI 

GRAPHICAL   SOLUTION   OF   A   SPHERICAL   TRIANGLE II5 

CHAPTER   XII 

RECAPITULATION   OF   FORMULAS Up 

APPENDIX 

RELATION   OF  THE  PLANE,  SPHERICAL,  AND   PSEUDO-SPHERICAL 

TRIGONOMETRIES 135 

ANSWERS  TO   EXERCISES 129 


PLANE  TRIGONOMETRY 

CHAPTER   I 
THE  TRIGONOMETRIC   FUNCTIONS 

ANGLES 

1,  In  Trigonometry  the  size  of  an  angle  is  measured  by 
the  amount  one  side  of  the  angle  has  revolved  from  the 
position  of  the  other  side  to  reach  its  final  position. 

Thus,  if  the  hand  of  a  clock  makes  one-fourth  of  a  rev- 
olution, the  angle  through  which  it  turns  is  one  right  angle ; 
if  it  makes  one-half  a  revolution,  the  angle  is  two  right  an- 
gles; if  one  revolution,  the  angle  is  four  right  angles;  if  one 
and  one-half  revolutions,  the  angle  is  six  right  angles,  etc. 


_B 


^-t5) 


FIG.  3 


The  amount  the  side  OB  has  rotated  from  OA  to  reach  its  final  position 
may  or  may  not  be  equal  to  the  inclination  of  the  lines.  In  Fig.  i  it  is  equal 
to  this  inclination  ;  in  Fig.  4  it  is  not. 

Two  angles  mny  have  the  same  sides  and  yet  be  different.     In  Fig.  2 

I 


PLANE    TRIGONOMETRY 

and  Fig.  4  the  posili(ins  of  the  sides  of  the  angles  are  the  same  ;  yet  in 
Fig.  2  the  angle  is  two  right  angles,  in  Fig.  4  il  is  six  right  angles.  Tlie 
addition  of  any  number  of  complete  revolutions  to  an  angle  does  not  change 
tlie  posi     >n  of  its  sides. 

Quti>tion. — Through  how  many  right  angles  does  the  hour-hand 
of  a  clock  revolve  in  6^  hours?  the  minute-hand  } 

Question. — If  the  fly-wheel  of  an  engine  makes  100  revolutions  per 
minute,  through  how  many  right  angles  does  it  revolve  in  i  second  } 


Initial  line 


1}    RIGHT   ANGLES 


Initial  line 


5I   RIGHT    ANGLES 


Dcf. — The  first  side  of  the  angle — that  is,  the  side  from 
which  the  revolution  is  measured — is  the  initial  line;  the 
second  side  is  the  terminal  line. 

Def. — If  the  direction  of  the  revolution  is  opposite  to  that 
of  the  hands  of  a  clock,  the  angle  is  positive;  if  the  same 
as  that  of  the  hands  of  a  clock,  the  angle  is  negative. 


Initial  line 


Initial  line 

POSITIVE   ANGLE 


NEGATIVE   ANGLE 


The  angles  we  have  employed  as  illustrations — those  described 

by  the  hands  of  a  clock — are  all  negative  angles. 

2,  Angles  are  usually  measured  in  degrees,  minutes,  and 
seconds.  A  degree  is  one-ninetieth  of  a  right  angle,  a  min- 
ute is  one-sixtieth  of  a  degree,  a  second  is  one-sixtieth  of  a 
minute. 


THE    TRIGONOMETRIC  FUNCTIONS  3 

The  symbols  indicating  degrees,  minutes,  and  seconds  are  ^  '  "; 
tlius,  twenty-six  degrees,  forty-three  minutes,  and  ten  seconds  is 
written  26°  43'  10". 

li.  The  plane  about  the  vertex  of  an  angle  is  div.  led  into 
four  quadrants,  as  shown  in  the  figure  ;  the  first  quadrant 
besfins  at  the  initial  line. 


Ill 


IV 


II 

III 

Initial  Line 
IV 

A 

»\ 

^\ 

■>\ 

"^<\, 

I 

■>  \ 

~~x 

A 

Initial  Line 

in 

IV 

THE  FOUR  QUADRANTS 


ANGLE  IN  1ST  QUADRANT 


ANGLE  IN  2D  QUADRANT 


II 

c 

I 

I 

\    Initial  Line 

III 

A IV 

W 

■>\ 

ANGLE  IN  3D  QUADRANT 


ANGLE  IN  4TH  QUADRANT 


An  angle  is  said  to  be  in  a  certain  quadrant  if  its  ternninal 
line  is  in  that  quadrant. 

EXERCISES 

4.  (I.)  Express  2\  right  angles  in  degrees,  minutes,  and  seconds. 
In  what  quadrant  is  the  angle  .-^ 

(2.)  What  angle  less  than  360°  has  the  same  initial  and  terminal 
lines  as  an  angle  of  745°? 

(3.)  What  positive  angles  less  than  720°  have  the  same  sides  as  an 
angle  of  —73°  ? 

(4.)  In  what  quadrant  is  an  angle  of  — 890°.' 


4  PLANE    TRIGONOMETRY 

DEFINITIONS   OF  THE   TRIGONOMETRIC  FUNCTIONS 
5,  The  trigonometric  functions  are  numbers,  and  are  de- 
fined as  the  ratios  of  lines. 

Let  the  angle  AOP  be  so  placed  that  the  initial  line  is 
horizontal,  and  from  P,  any  point  of  the  terminal  line,  draw 
PS  perpendicular  to  the  initial  line. 


^    lt<i5 


S     A 


ANGLE  IN  THE  1ST  QUADRANT 


ANGLE  IN  THE  2D  QUADRANT 


^a? 

®    r 

^ 

y" 

0 

y^ 
/p 

ANGLE  IN  THE  3D  QUADRAHT 


ANGLE  IN  THE  4TH  QUADRANT 


Denote  the  angle  AOP  by  x. 


SP        ^^'^ 

-— ^=sine  of  x  (written  sin;r). 

OS 

— -  =coslne  of  x  (written,  cos;ir). 


THE    TRIGONOMETRIC  FUNCTIONS 


SP 

-y^  :=  tangent  of  x  (written  tan;r).  ' 


cotangent  of  x  (written  cot;tr). 


sp'' 

OP 

-y-.  =  secant  of  x  (writteq  secjr). 


OP 
SP 


—  cosecant  of  x  (writte 


To  the  above  may  be  added  the  versed  sine  (written  versin)  and  coversed 
sine  (written  coversin),  which  are  defined  as  follows : 

versin  ic  =  i  —  cos  x\  coversin  x  =  i  —  sin  x,. 

Tlie  values  of  the  sine,  cosine,  etc.,  do  not  depend  upon 

what  point  of  the  terminal  line  is  taken  as  P,  but  upon  the 

angle. 


^, 

..-' 

^  p^ 

/^ 

o 

S  i 

> 

s's 


For  the  triangles  OSP  and  OS F  being  similar,  the  ratio  of  any 
two  sides  of  OS' P'  is  equal  to  the  ratio  of  the  corresponding  sides 
of  OSP. 

Def. — The  sine,  cosine,  tangent,  cotangent,  secant,  and 
cosecant  of  an  angle  are  the  trigonometric  functions 
of  the  angle,  and  depend  for  their  value  on  the  angle 
alone. 

G,  A  line  may  by  its  length  and  direction  represent  a 
number;  the  magnitude  of  the  number  is  expressed  by  the 
IcngtJi  of  the  line;  the  number  xs  positive  or  negative  ac- 
cordinrj  to  the  direction  of  the  line. 


6  PLANE   TRIGONOMETRY 

7.  In  §  5,  if  the  denominators  of  the  several  ratios  be 
taken  equal  to  unity,  ^/le  trigonometric  functions  will  be  rep- 
resented by  lines. 


Thus,  sin^: 


OP' 


SP 

—  =  SP— the  number  represented  by 

the  line,  that  is,  the  ratio  of  the  line  to  its  unit  of  length. 

Hence  SP  may  represent  the  sine  of  x. 

In  a  similar  manner  the  other  trigonometric  functions 
may  be  represented  by  lines. 

In  the  following  figures  a  circle  of  unit  radius  is  described 
about  the  vertex  O  of  the  angle  A  OP,  this  angle  being  de- 
noted by  X.     Then  from  §  5  it  follows  that 


FIG.  3 


FIG  4 


THE   TRIGONOMETRIC  FUNCTIONS  "V/ 

SP  represents  the  sine  of  x. 
OS  represents  the  cosine  of  x. 
A  T  represents  the  tangent  of  x. 
BC  represents  the  cotangent  of  x. 
(97^  represents  the  secant  of  x. 
OC  represents  the  cosecant  of  ;r. 

For  the  sake  of  brevity,  the  lines  SP,  OS,  etc.,  of  the  preceding  figures  are 
often  spoken  of  as  the  sine,  cosine,  etc. 

Hence,  we  may  also  define  the  trigonometric  functions 
in  general  terms  as  follows: 

If  a  circle  of  unit  radius  is  described  about  the  vertex  of 
an  angle, 

(I.)  The  sine  of  the  angle  is  represented  by  the  perpendicular 
upon  the  initial  line  from  the  intersection  of  the  terminal  line  with 
the  circumference. 

(2.)  The  cosine  of  the  angle  is  represented  by  the  segment  of  the 
initial  line  extending  from  the  vertex  to  the  sine. 

(3.)  The  tangent  of  the  angle  is  represented  by  a  line  tangent  to 
the  circle  at  the  beginning  of  the  first  quadrant,  and  extending  from 
the  point  of  tangency  to  the  terminal  line. 

(4.)  The  cotangent  of  the  angle  is  represented  by  a  line  tangent 
to  the  circle  at  the  beginning  of  the  second  quadrant,  and  extending 
from  the  point  of  tangency  to  the  terminal  line. 

(5.)  The  secant  of  the  angle  is  represented  by  the  segment  of  the 
terminal  line  extending  from  the  vertex  to  the  tangent. 

(6.)  The  cosecant  of  the  angle  is  represented  by  the  segment  of 
the  terminal  line  extending  from  the  vertex  to  the  cotangent. 

The  definitions  in  §  5  are  called  the  ratio  definitions  of  the  trigonometric 
functions,  and  those  in  §  7  the  line  definitions.  The  introduction  of  two 
definitions  for  the  same  thing  should  not  embarrass  the  student.  We  have 
shown  that  they  are  equivalent.  In  some  cases  it  is  convenient  to  use  the 
first  definition,  and  in  other  cases  the  second,  as  the  student  will  observe 
in  the  course  of  this  study.  It  is  therefore  important  that  he  should  be- 
come familiar  with  the  use  of  both. 


8 


PLANE   TRIGONOMETRY 


SIGNS   OF   THE   TRIGONOMETRIC   FUNCTIONS 

S,  Lines  are  regarded  as  positive  or  negative  according 
to  their  directions.  Thus,  in  the  figures  of  §  5,  OS  '\s  posi- 
tive if  it  extends  to  the  right  of  O  along  the  initial  line, 
negative  if  it  extends  to  the  left ;  SP  \^  positive  if  it  extends 
iipivnrd  from  OA,  negative  if  it  extends  downward.  OP,  the 
terminal  line,  is  dXwd^y?, positive. 

The  above  determines,  from  §  5,  the  signs  of  the  trigono- 
metric functions,  since  it  shows  the  signs  of  the  two  terms 
of  each  ratio. 

By  the  line  definitions  the  signs  may  be  determined  di- 
rectly. The  sine  and  tangent  are  positive  if  measured  up- 
ward from  OA,  and  negative  if  measured  doivnward. 

The  cosine  and  cotangent  are  positive  if  measured  to  the 
right  from  OB,  and  negative  if  measured  to  the  left, 

B       cot+  Cot-   B      --^- 


p> 

Co«-\ 

^ 

1      5 

5        0 

N 

nc.  4 


THE   TRIGONOMETRIC  FUNCTIONS 


The  secant  and  cosecajit  are  positive  if  measured  in  the 
same  direction  as  the  terminal  line,  OP;  negative  if  measured 
in  the  opposite  direction. 

The  signs  of  the  functions  of  angles  in  the  difTerent  quadrants  are  as  follows  : 


Quadrant 

I 

II 

Ill 

IV 

Sine  and  cosecant 

+ 

+ 

- 

Cosine  and  secant 

+ 

- 

- 

+ 

Tangent  and  cotangent 

+ 

- 

+ 

- 

9.  It  is  evident  that  the  values  of  the  functions  of  an 
angle  depend  only  upon  the  position  of  the  sides  of  the 
angle.  If  two  angles  differ  by  360°,  or  any  multiple  of  360°, 
the  position  of  the  sides  is  the  same,  hence  the  values  of 
the  functions  are  the  same. 


Thus  in  Fig.  i  the  angle  is  120°  in  Fig.  2  the  angle  is  840°   yet 
the  lines  which  represent  the  functions  are  the  same  for  both  angles. 

EXERCISE 
Determine,  by  drawing  the  necessary  figures,  the  sign  of  tan  1000°; 
cos  810°;  sin  760°;  cot  —70°;  cos  — 550°;  tan  —560°;  sec  300°;  cot 
1560°;  sin  130°;  cos  260°;  tan  310°. 


lO 


PLANE   TRIGONOMETRY 


RELATIONS  OF  THE  FUNCTIONS 

10,  By  §  5,  whatever  may  be  the  length  of  OP,  we  have 

SP       .  OS  SP     ^  OS        ^       OP  OP 

=  sm  jr  :  =  cos  x  ;  — —  =  tan  x  ;  -—  =  cot  jr ;  — —  =  sec  x  ;  — -  =  esc  x. 

or  '  OP  OS  SP  OS  SP 


We  have,  then,  from  Figs.  2  and  3, 


SP     ^  sin  a? 

ttt;  =  tan  X  = ; 

OS  COS  a;' 

OS  ^         cos  05 

SP  sin  a; 


Multiplying  (i)  by  (2), 


or 


tana?  cot 00=1, 

tanjr  = ;     cot;r  =  - 

cot  x  tan  x 


FIG.  3 


Again,  from  Figs.  2  and  3, 

OP                    1 
•TT77  =  sec  X  = ; 

OS  COS  X  » 

OP  _  1 

__C8ca?-g^^j^. 

From  Figs.  2  and  3,  OS'  +  SP'=OP\ 
or  sln'aj  +  co8*ar  =  1, 

and  sin*jr=  I  —  cos'jr ;     cos'jr=  i  —  sin*jr. 

Also,  OA'+Ar=Or,  and  OB'  +  BC'  =  OC' 
or  ^  +  tan'a?  =  sec*  a? ; 

1  +  cot'a?  =  C8c*a5. 


0) 

(2) 

(3) 

(4) 
(5) 

(6) 


(7) 
(8) 


THE   TRIGONOMETRIC  FUNCTIONS  ii 

The  angle  x  has  been  taken  in  the  first  quadrant ;  the 
results  are,  however,  true  for  any  angle.  The  proof  is  the 
same  for  angles  in  other  quadrants,  except  that  SP  be- 
comes negative  in  the  third  and  fourth  quadrants,  and  OS 
in  the  second  and  third. 

EXERCISES 
Urn  (I.)  Prove  cos;r  sec.rzr  I. 

(2.)  Prove  sin^  csc,r==i. 
(3.)  Prove  tan. r  cos ;r  =  sin  jr. 


(4.)  Prove  sin  .r  \  i  —  cos'^  ;r  =  i  —  cos' x. 
(5.)  Prove  tan  jir  +  cot  .r  =  ■ 


sin.r  cos^ 
(6.)  Prove  sin* X  — cos* .r=  1  —  2  cos* jr. 

(7.)  Prove =  sin  jr. 

cotjr  secjr 

(8,)  Prove  tan  x  sin  ,r  -|-  cos  jr  =  sec  jr. 

1^,  The  formulas  (i)-(8)  of  §  10  are  algebraic  equations 
connecting  the  different  functions  of  the  same  angle.  If 
the  value  of  one  of  the  functions  of  an  angle  is  given,  we 
can  substitute  this  value  in  one  of  the  equations  and  solve 
to  find  another  of  the  functions.  Repeating  the  process,  we 
find  a  third  function,  etc. 

In  solving  equation  (6),  (7),  or  (8)  a  square  root  is  extracted ; 
unless  something  is  given  which  determines  whether  to  choose  the 
positive  or  negative  square  root,  we  get  two  values  for  some  of 
the  functions.  The  reason  for  this  is  that  there  are  two  angles 
less  than  360^  for  which  a  function  has  a  given  value. 

EXERCISES 

13.  (I.)  Given  x  less  than  90°  and  sinjr  =  ^;  find  all  the  other 

functions  of  jr. 

Solution. — 

cosjr=:  ±  v^  1  — i=  ±4\/3. 

Since  x  is  less  than  90^,  we  know  that  cosjr  is  positive. 


12  PLANE   TRIGONOMETRY 

Hence  oosjc  =  +  \V'i  ; 

tanjr=— -r;  =  iv^3'; 

cotjr  = =v  •? : 

I  _ 

sec  jr  =  — —  =.\y/  "W 
K3 

I 
cscj:  =  -  =  2. 

2 

(2.)  Given  tan;r  =  — ^  and  x  in  quadrant  IV;  find  sin^  and  cos  jr. 

Solution. — 

sin  X  _       . 

COSJT 

hence  3  sin  jt  =  —  cos  j:, 

sin*  jc  +  cos*  jf  =  I ; 
hence  lo  sin*jr  =  i  ; 

sin  JT  =  -  a/ ^=  -  j^Vio; 

COSX=/jV/lO. 

(3.)  Given  sin( — 30°)  =  — -J;  find  the  other  functions  of — 30*^. 

(4.)  Given  x  in  quadrant  III  and  sin:r  =  — i;  find  all  the  other 
functions  of  x. 

(5.)  Give"\i  y  in  quadrant  IV  and  sin^=  —  |,  find  all  the  other 
functions  of  ^. 

(6.)  Given  cos  60°  =  J ;  find  all  the  other  functions  of  60°. 

(7.)  Given  sin  o°  =  o;  find  cos  0°  and  tanoo, 

(8.)  Given  tan^  =  Jand  z  in  quadrant  I;  find  the  other  functions 
of  ^. 

(9.)  Given  cot45°=  i ;  find  all  the  other  functions  of  45°. 

(10.)  Given  tanj=^-v/5  and  cos^'  negfative;  find  all  the  other 
functions  o{ y. 

(II.)  Given  cot 30°=  ^^3;  find  the  other  functions  of  30°. 

(12.)  Given  2  sin;r  =  i — cosjr  and  x  in  quadrant  II;  find  sinjr 
and  cos  jr. 

(13.)  Given  tan  j"-|-cotjr  =  3  and  jr  in  quadrant  I ;  find  sin  jr. 


THE   TRIGONOMETRIC  FUNCTIONS 


'3 


FUNCTIONS   OF  AN   ACUTE   ANGLE   OF  A   RIGHT   TRIANGLE 

14:,  The  functions  of  an  acute  angle  of  a  right  triangle 
can  be  expressed  as  ratios  of  the  sides  of  the  triangle. 


Remark. — Triangles  are  usually  lettered,  as  in  Fig.  2,  the  capital 
letters  denoting  the  angles,  the  corresponding  small  letters  the  sides 
opposite. 

In  the  right  triangle  ABC,  by  §  5, 

^      .      BC     a  ' 

sIn-4  =  — — =  -  =cos5; 
AB     c 

^      AC     h       .    „ 
cos  A  =  ——1  =  -  =  sin  5 ; 
AB       C 

.      BC     a  „ 

tan  ^  = -— ,=  ^  =  cot  ^ ; 
AC      0 

.    .      AC     b 
cotA=—  =  -=t^nB. 

15,  From  §  14,  for  an  acute  angle  of  a  right  triangle,  we  have 

side  opposite  angle 

sine  = ,   '^^ ^:^— ; 

hypotenuse 

side  adjacent  to  ansrle  ' 

cosine  = r-i 2— ; 

hypotenuse 


tangent : 


side  opposite  angle 
side  adjacent  to  angle' 

side  adjacent  to  angle 

cotangent  = — r^ — -. — ~— ■ 

°  side  opposite  angle 


(9) 


14  PLANE   TRIGONOMETRY 


FUNCTIONS  OF  COMPLEMENTARY  ANGLES 

IQ,  From  §  14,  we  have 

sin  ^=008  5=008(9©"— /I);  1 
cos  ^=  sin  5=  sin  (90° -4);  I 
f  an  ^  =  cot  5 = cot  (90° — ^) ; 
cot  ^  =  tan  5  =  Ian  (90"  -  ^). 

Because  of  this  relation  the  sine  and  cosine  are  called  co-func- 
tions of  each  other,  and  the  tangent  and  cotangent  are  called  co- 
functions  of  each  other. 

The  results  of  this  article  may  be  stated  thus: 
A  function  of  an  acute  angle  is  equal  to  the  co-function  of 
its  cofnplementary  angle. 

The  values  of  the  functions  of  the  different  angles  are  given  in  "  Trigo- 
nometric Tables."  By  the  use  of  the  principle  just  proved,  each  function 
of  an  angle  between  45"  and  90°  can  be  found  as  a  function  of  an  angle  less 
than  45°.  Consequently,  the  tables  need  to  be  constructed  for  angles  up  to 
45°  only.  The  tables  are  so  arranged  that  a  number  in  them  can  be  read 
either  as  a  function  of  an  angle  less  than  45°  or  as  the  co-function  of  the 
complement  of  this  angle. 

EXERCISES 

ly,  (I.)  Express  as  functions  of  an  angle  less  than  45°: 
sin  70° ;  cos  89°  30' ;         tan  63° ; 

cos 66° ;  cot 47° ;  sin  72°  39'. 

(2.)  cos-r  =  sin2jr;  find  .r. 
(3.)  tan  X  =  cot  yc ;  find  x. 
(~~  (4.)  sin2Jr  =  cos3Jr;  find  x. 

(5.)  cot(30°— jr)  =  tan(30°-(-3.r);  find  ;r. 

(6.)  A,  B,  and   C  are  the   angles  of  a  triangle;  prove  that 
cos\B=zsm\{A-\-C). 
Hint.—  A+B+C=i8d>. 


THE    TRIGONOMETRIC  FUNCTIONS 


15 


FUNCTIONS   OF  O^,  90",  l8o°,  270°,  AND  360° 

IS.  As  the   angle  x  decreases  towards  0°  (Fig.  i),  sin^  de- 
creases and  cosx  increases.     When  OP  comes  into  coincidence 
with  OA,  ^-P  becomes  o,  and  (96' becomes  0A{  —  \). 
Hence  sino°=:o.     coso°=i. 


B 

c 

V\ 

/ 

^ 

^::^^^ 



0 

A 

FIG.  3 


As  the  angle  x  increases  towards  90°  (Fig.  2),  sinx  increases 
and  co%x  decreases.     When  (9/' comes  into  coincidence  with  OB, 
6"/* becomes  OB{~i)  and  t?.^?  becomes  o. 
Hence  sin9o°  =  i,     cos9o°=^o. 

As  the  angle  x  decreases  towards  0°  (Fig.  3),  tanx  decreases 
and  cot^  increases.     When  (9/' comes  into  coincidence  with  OA, 
^/"becomes  o  and  ^C  has  increased  without  limit. 
Hence  tano°  =  o,     cot  0°  =  00, 

As  the  angle  x  increases  towards  90°  (Fig.  4),  tan.r  increases 
and  cot^  decreases.     When  OP  comts  into  coincidence  with  OB, 
.^7"  has  increased  without  limit,  and  BC=o. 
Hence  tan  90°  =  00,     cot  90°  =  0. 

Remark. — By  coto°=ao  we  mean  that  as  the  angle  approaches  indefinitely 
near  to  0°  its  cotangent  increases  so  as  to  become  greater  than  any  finite  quan- 
tity we  may  choose.  The  symbol  00  does  not  denote  a  definite  number,  but 
simply  that  the  number  is  indefinitely  great. 


i6 


PLANE   TRIGONOMETRY 


In  every  case  where  a  trigonometric  function  becomes  indefinitely 
great  it  is  in  a  positive  sense  if  the  angle  approaches  the  limiting 
value  from  one  side,  in  a  negative  sense  if  the  angle  approaches  the 
limiting  value  from  the  other  side.  Thus  cot  0°= +  00  if  the  angle 
decreases  to  0°  but  cot  0°=  — 00  if  the  angle  increases  from  a  nega- 
tive angle  to  0°.  We  shall  not  often  need  to  distinguish  between 
-}-oo  and  —00,  and  shall  in  general  denote  either  by  the  symbol  00. 

By  a  similar  method  the  functions  of  180°,  270°,  and  360°  may  be 
deduced.     The  resuhs  of  this  article  are  shown  in  the  following  table : 


Angle 
sin 

o"^ 

90° 

180° 

270° 

360° 

0 

I 

0 

—  I 

0 

cos 

I 

0 

—  I 

0 

I 

tan 

0 

CO 

0 

CO 

0 

cot 

00 

0 

—  00 

0 

00 

Ifi,  It  may  now  be  stated  that,  as  an  angle  varies,  its  sine  and  cosine 
can  take  on  values  from  —  i  to  -\-  r  only,  its  tangent  and  cotangent  all 
values  from  —  so  A;  -f  00 ,  its  secant  and  cosecant  all  values  from  —  00 
/(?-(-  00 ,  ex.  ]pt  those  between  —  i  and  -|-  /. 


FUNCTIONS  OF  THE  SUPPLEMENT  OF  AN  ANGLE 

20,  Suppose  the  triangle  OPS  (Fig.  i)  equal  to  the  tri- 
angle  OP'S'  (Fig.  2),  then  SP=S'P'  and  OS^  OS',  and  the 
angle  AOP'  (Fig.  2)  is  equal  to  the  supplement  of  AOP 
(Fig.  i).  Also,  in  the  triangle  AOP'  (Fig.  3),  angle  AOP' 
=  c\r\<^\Q  AOP'  {Y\g.  2). 


PIG.  3 


THE   TRIGONOMETRIC  FUNCTIONS 


17 


(10) 


It  follows  from  §§  5  and  8  that 

sin  (1§0°  —  a?)  =  8ln  ic ;       1 
cos  (180°  —  a?)  =  —  cos  OR ; 
tan  (1§0°  —  ic)  =  —  tan  x ; 
cot  (1§0° —  a?)  =  —  cot  x. 

The  results  of  this  article  may  be  stated  thus : 

The  sine  of  an  angle  is  eqnal  to  the  sine  of  its  supplement, 

and  the  cosine,  tangent,  and  cotangent  are  each  equal  to  minus 

the  same  functions  of  its  supplement. 

The  principle  just  proved  is  of  great  importance  in  the  solution  of  tri- 
angles which  contain  an  obtuse  angle. 

FUNCTIONS   OF  45°,    30°,   AND   6o° 

21.  In  the  right  triangle  OSP  (Fig.  i)  angle  (9  =  angle  P  =  AS°' 
and  OP  =  i. 

Hence  OS=SP  =  ^  ^2. 

Therefore  sin45°  =  cos45°=^-v/2;  §§  14,  16 

tan  45°  =  cot  45°=  I. 


i     s 


In  equilateral  triangle  OP  A  (Fig.  2)  the  sides  are  of  unit  length 
PS  bisects  angle  OP  A,  is  perpendicular  to  OA,  and  bisects  OA. 
Hence,  in  the  right  triangle  OPS,  0S  =  ^,  SP  =  ^y/i. 
Therefore  sin  30°r3COs6o°  =  l;  §14 

cos  30°  =  sin  60°  =^^3; 
tan  30°  =  cot  60°  =  ^  V'3 ; 
cot  30°  =  tan  60°  =  v'3. 
2 


i8  PLANE   TRIGONOMETRY 

22.  The  following  values  should  be  remembered : 


Angle 

0° 

30° 

45° 

60° 

90° 

sin 

o 

i 

iv/2 

i\/3 

I 

cos 

I 

iV3 

iV2 

i 

0 

EXERCISES 

Prove  that  if  x  =  30°, 

(I.)  sin  2j:  =  2  sinjjr  cos;r; 

(2.)  cos  3^  =  4  cos*  x  —  2,  cos  X ; 

(3.)  cos  2;r  =  cos'';r  —  sin' 4:; 

(4.)  sin  3;r=  3  sinjr  cos'^  —  sin'jr; 

2  tan  X 

(5.)  tan2;tr  = 7—. 

^•"  I— tan-jr 

(6.)  Prove  that  the  equations  of  exercises  i  and  3  are  cor- 
rect if  ;ir  =  45°. 

(7)  Prove  that  the  equations  of  exercises  (2)  and  (4)  are  cor- 
rect if  ;t-=  120°. 


The  following  three  articles,  §§  23-25,  are  inserted  for 
completeness.  They  include  the  functions  of  (90 — x)  and 
(180— ;r),  which,  on  account  of  their  great  innportance,  were 
treated  separately  in  §§  16  and  20. 


FUNCTIONS  OF  {—x),  (l8o°— ;ir),  (l8o°4-;»^),  (360°  — ;ir) 

23.  The  line  representing  any  function — as  sine,  cosine,  etc. 
— of  each  of  these  angles  has  the  same  length  as  the  line  repre- 
senting the  same  function  of  x. 

Thus  in  Figs.  2  and  3,  triangle  (?57'=: triangle  OSP,  hence  SP=S'P', 
and  OS=OS'. 


THE    TRIGONOMETRIC  FUNCTIONS 
B  C         C'  B 


19 


B 

c 

/ 

/^^ 

^^NC/ 

T 

/ 

4- 
3     f 

\x 

^ 

A 

V 

/ 

0 

'1 

p 

v_ 

/ 

^\P^ 

"~\P/ 

f 

\ 

N^ 

-ae 

\x 

\ 

A 

s' 

v 

0 

\ 

x^i 

\ 

V 

> 

< 

t' 

FIG.    2 

B 


XX' 

^\p/ 

T 

(T 

\ 

A 

1            ^ 

S 

/ 

v_ 

^^p^ 

t' 

In  Figs.  I  and  4,  triangle  OSF' =tnang\e  OSP.  hence  SP'=SP. 

In  Figs.  I,  2,  and  4,  triangle  OA  7"=triangle  OA  T,  hence  A  T'  =  A  7. 

In  Figs.  I,  2,  and  4,  triangle  C>j9C"  =  triangle  O^C.  hence  i?C"'=.gC. 

Therefore  any  function  of  each  of  the  angles  {  —  x),  (180''— jc), 
(i  80°  +  x),  (360°  —x),  is  equal  in  numerical  value  to  the  same  function 
of  X.  Us  sign,  however,  defends  oji  the  direction  of  the  line  repre- 
senting it. 

Putting  in  the  correct  sign,  we  obtain  the  following  table; 


sin  ( —  jr)  =  —  sin  x 
cos(  — x)  =  cos  J? 
tan  { —  jr)  =  —  tan  x 
cot(— jt)  =  —  cotjf 

sin  (i8o°4-jr)=  —  sinj; 
cos  ( 1 80°  +  jt)  =  —  cos  J? 
tan  C 1 80°  -t-  jr)  =  tan  x 
cot(i8oO  + jr)  =  cot.r 


sin  (180°  —  jr)  =  sin  j; 
cos  (180°  —  jr)  =  —  cos  jr 
tan  (180°  —  jr)=  —  tanj; 
cot  (i  80°  —  x)  =  -  cot Jf 

sin  (360°  —  x)  =  —  sin  j; 
cos  (360''  —  Jf)  =  cosx 
tan  (360°  —  x)=.  —  tan  x 
cot  (360°  —  x)=.  —  cot  x 


PLANE   TRIGONOMETRY 


FUNCTIONS   OF  (90°— j),  (90° +j),  (270° -j),  (270° +j) 

^4.  The  line  representing  the  sine  of  each  of  these  angles  is 
of  the  same  length  as  the  line  representing  the  cosine  o\  y,  the 
cosine,  tangent,  or  cotangent,  respectively,  are  of  the  snme  length 
as  the  sine,  cotangent,  and  tangent  oi y. 

T 


FIG.  3 


For 


Triangle  05'/' =  triangle  OSP,  hence  S' P'  =  OS,  and  05'  =  SP. 
Triangle  OA  T  —  triangle  OBC,  hence  A  T'  =  RC. 
Triangle  OBC  =  triangle  OA  T,  hencf  BC  =  AT. 

Therefore  any  function  of  each  of  the  angles  (90°  —y),  (90°  +>'), 
(270°—^),  (270° +_y),  is  equal  in  numerical  value  to  the  co-function 


THE    TRIGONOMETRIC  FUNCTIONS  21 

of  y.     Its  sign,  however,  depends  on  the  direction  of  the  line  repre- 
senting it. 

Putting  in  the  correct  sign,  we  obtain  the  following  table  : 

sin  (90°  —  v)  =  cos^  sin  (90°  +  y)  =  cos^ 

cos  (90°  —  v)  =  sin  v  cos  (90°  +  j-)  =  —  smy 

tan  (90°  —  y)  =  coty  tan  (90°  +  ;')  =  —  cot  v 

cot  {90°  —y)  =  tan  J  cot  (90°  +i')  =  —  tan^' 

sin  (270°  —  ;')  =  —  cosv  sin  (270°  +y)  =  —  cosy 

cos  (270°  —  y)  =  —  sin  J  cos  (270°  +  ^)  =  sin/ 

tan  (270°  —  1)  =  coty  tan  (270°  +  y)  =  —  cot>' 

cot  (270°  —  r)  =:  tanj'  cot  (270°  +1)  =  —  tan  v 

2S,  Either  of  the  two  preceding  articles  enables  us  directly  to 
express  the  functions  of  any  angle,  positive  or  negative,  in  terms 
of  the  functions  of  a  positive  angle  less  than  90°. 

Thus,  sin  212°  =sin  (180°  +  32°)  =  —  sin  32° ; 

cos  260°  =  cos  (270°— 10°)  =  —  sin  10°. 


EXERCISES 

(I.)  What  angles  less  than  360°  have  the  sine  equal  to  —  J\/2  ?  the 
tangent  equal  tov^3  ' 

(2.)  For  what  values  of  x  less  than  720°  is  sin.r  =  -^l/2? 

(3.)  Find  the  sine  and  cosine  of  —30°;  765°;  120°;  210°. 

(4.)  Find  the  functions  of  405°;  600^;  1125°;  —45°;  225°. 

(5.)  Find  the  functions  of  —120°;  —225°;  —420°;  3270°. 

(6.)  Express  as  functions  of  an  angle  less  than  45°  the  functions  of 
233°;  —  197°;  894°. 

(7.)  Express  as  functions  of  an  angle  between  45°  and  90^,  sin  267° ; 
tan  (  —  254°) ;  cospso'^. 

(8.)  Given  cos  164°  =  —  .96,  find  sin  196°. 

(9.)  Simplify  cos(9o°-(--*')cos(27o°  — ;r)  — sin(i8o°  — ;r)sin(36o°  — .r). 

^      ^  „.       ...   sin(i8oo  — j:)»      ,     n  ,      ^  ,  ' 

(10.)  Simplify— tan(90°4--*')4 


sin  (270°  —  x)       ^y^    ^  ■        sin*  (270°  — .r) 
(II.)  Express  the  functions  of  (jr  — 90'^)  in  terms  of  functions  of  x. 


CHAPTER   II 


THE  RIGHT  TRIANGLE 

27»  To  solve  a  triangle  is  to  find  the  parts  not  given. 

A  triangle  can  be  solved  if  three  parts,  at  least  one  of 
which  is  a  side,  are  given.  A  right  triangle  has  one  angle, 
the  right  angle,  always  given  ;  hence  a  right  triangle  can 
be  solved  if  two  sides,  or  one  side  and  an  acute  angle,  are 
also  given. 

The  parts  of  the  right  triangle  not  given  are  found  by 
the  use  of  the  following  formulas: 

adjacent  side 


,  ,    .  opposite  side 

(I)  sine       =-p^ 

hypotenuse 


(3)  tangent  = 


opposite  side 


(2)  cosine        = 
(4)  cotangent  = 


hypotenuse 

adjacent  side 

opposite  side 


14 


adjacent  side  ' 

(5)  c^=a^-\-i>';  (6)  B  =  {90°—A).  §  16 

To  solve,  select  a  formula  in  which  two  given  parts  enter;  substituting 
in  this  the  given  values,  a  third  part  is  found.  Continue  this  method  till 
all  the  parts  are  found. 

In  a  given  problem  there  are  several  ways  of  solving  the  triangle ;  choose 
the  shortest. 

EXAMPLE 

The  hypotenuse  of  a  right  triangle  is  47.653,  a  side  is 
21.34;  find  the  remaining  parts  and  the  area. 

B 


THE   RIGHT   TRIANGLE 


23 


SOLUTION   WITHOUT   LOGARITHMS 
The  functions  of  angles  are  given 
in  the  table  of  "  Natural  Functions." 


%\VlA  =-= 


21.34 


"47-653 
47.653)21.3400(^4478 
1906 I 2 

227880 
igo6i2 


372680 
333571 
39x090 
381224 
9866 

sin  .4  =  .4478 
^=26^"  36' 

b-=^c  cos  A 
=47. 653  X. 8942 

47-653 
.8942 


9530b 
190612 
428877 
381224 
42.6113126 
^=42.61  f 

^  =  (90''- 26°  36  |rr63°  24 

area  =^(7/^ 

=  ^x  21.34x42.61 

21.34 
42.61 

2134 
12804 

4268 

8536 


2)909.2974 
454.6487 
area=454.6 


SOLUTION   EMPLOYING  LOGARITHMS 

It  is  usually  better  to  solve  triangles 
by  the  use  of  logarithms. 

The  logarithms  of  the  functions  are 
given  in  the  tables  of  "  Logarithms  of 
Functions."  * 

.      .     a 

sin  W=:  — 

c 

log  sin  /4  =log  a — log  c 

log2i.34  =1.32919 

log  47-653=  I- 67809 

— sub. 

log  sin  .,4 =9.65  no—  ro 

^=26°  36'  14" 


cos  A=  - 

c 

log  ^=log  r +log  cos  A 
log  47- 653  =  1.67809 
log  cos  26°  36'  14"  =9.95140— 10 
log /^=  1.62949 

(5=42.6o8 


^=(90= -26°  36'  I4")=63°  23'  46" 

area  =  \al) 
log  area = log  |  +  log  a  +  log  /; 

log  1=9.69897 -10 
log2i. 34=1. 32919 
log  42  608  =  1 . 62949 
log  area=2. 65765    , 

area  =  454.62 


*  In  this  solution  the  five-place  table  of  the  "  Logarithms  of  Functions"  is 
used. 

\  No  more  decimal  places  are  retained,  because  the  figures  in  them  are  not 
cccur.ate  ,  thi.s  is  due  to  the  fnct  tliat  the  table  of  "  Natural  Functions"  is  only 
lour-place. 


24  PLANE    TRIGONOMETRY 

CHECK   ON   THE   CORRECTNESS   OK   THE   WORK 
d'  =  c''  -b''  =  {c^b){c-b)  1 

=  90.263  X  5.043 

90.263 
5043 


270789 
361052 
451315O 

«'  =  455- 196309 
Extracting   the   square    root,   a  = 
21.34,  which  proves  the  solution  cor- 
rect. 


a'' =  c^  -  b^  =  (c  +  b){c -•  fi) 
=  90.261  X  5.045 


log  90.261  =  1.95550 
log   5.045  =  0.70286 
2)2.65836 
log  21.34=  1. 32918 
fl  =  21.34,  which  proves  the  solu- 
tion correct. 


Reward. — The  results  obtained  in  the  solution  of  the  preceding 
exercise  without  logarithms  are  less  accurate  than  those  obtained  in 
the  solution  by  the  use  of  logarithms;  the  cause  of  this  is  that  four- 
place  tables  have  been  used  in  the  former  method,  five  place  in  the 
latter. 

EXERCISES 

28.  (1.)  In  a  right  triangle  ^  =  96.42,  ^=  1 14.81 ;  find  a  and  ^. 

(2.)  The  hypotenuse  of  a  right  triangle  is  28.453,3  side  is  18.197; 
find  the  remaining  parts. 

(3.)  Given  the  hypotenuse  of  a  right  triangle  =  747.24,  an  acute 
angle  =23°  45' ;  find  the  remaining  parts. 

(4.)  Given  a  side  of  a  right  triangle  =  37.234,  the  angle  opposite 
=  54°  27' ;  find  the  remaining  parts  and  the  area. 

(5.)  Given  a  side  of  a  right  triangle  =  1. 1293,  the  angle  adjacent 
=  74°  13'  27" ;  find  the  remaining  parts  and  the  area. 

(6.)  In  a  right  triangle  ^=:  15°  22' 11",  <r  =  . 01793;  find  <5. 

(7.)  In  a  right  triangle  5  =  71°  34' 53",  (^  =  896.33;  find  a. 

(8.)  In  a  right  triangle  r  =  3729.4.  ^  =  2869.1 ;  find  A. 

(9.)  In  a  right  triangle  a=  1247,  3=  1988  ;  find  c. 

CO.)  In  a  right  triangle  a  =  8.6432,  (5  =  4.7815  ;  find  Z>. 

The  angle  of  elevation  or  depression  of  an  object  is  the 
angle  a  line  from  the  point  of  observation  to  the  object 
makes  vi^ith  the  horizontal. 


THE  RIGHT    TRIANGLE  25 


^ITus  angle  x  (Fig.  i)  is  the  angle  of  elevation  oi  P  \l  O  is  the  point  of 
observation  ;  angle  y  (Fig.  2)  is  the  angle  of  depression  oi  P  \l  O  is  the 
point  of  observation. 

(II.)  At  a  horizontal  distance  of  253  ft.  from  the  base  of  a  tower  the 
angle  of  elevation^of  the  top  is  60°  20' ;  find  the  height  of  the  tower. 

(12.)  Fron*'t|je  top  of  a  vertical  cliff  85  ft.  high  the  angle  of  depres- 
sion of  a  buoy  is  24°  31'  22";  find  the  distance  of  the  buoy  from  the 
foot  of  the  cliff. 

(13.)  A  vertical  pole  31  ft.  high  casts  a  horizontal  ^^h^dow  45  ft.  long ; 
find  the  angle  of  elevation  of  the  sun  above  the  horizbn. 

(14.)  From  the  top  of  a  tower  115  ft.  high  the  angle  of  depression 
of  an  object  on  a  level  road  leading  away  from  the  tower  is  22°  13'  44" ; 
find  the  distance  of  the  object  from  the  top  of  the  tower. 

(15.)  A  rope  324  ft.  long  is  attached  to  the  top  of  a  building,  and 
the  inclination  of  the  rope  to  the  horizontal,  when  taut,  is  observed 
to  be  47^"  21'  17";  find  the  height  of  the  building. 

(16.)  A  light-house  is  150  ft.  high.  How  far  is  an  object  on  the 
surface  of  the  water  visible  from  the  top  ? 

[Take  the  radius  of  the  earth  as  3960  miles.] 

(17.)  Three  buoys  are  at  the  vertices  of  a  right  triangle;  one  side 
of  the  triangle  is  17,894  ft.,  the  angle  adjacent  to  it  is  57*^  23'  46".    .. 
Find  the  length  of  a  course  around  the  three  buoys.    ('V^OkX\W»<\)'>"^*} 

(18.)  The  angle  of  elevation  of  the  top  of  a  tower  observed  from  a 
point  at  a  horizontal  d\«tat«;e  of  897.3  ft.  from  the  base  is  10°  27'  42"; 
finc^  the  height  of  the  tower. 

"^ft9.)  A  ladder  42i  ft.  long  leans  against  the  side  of  a  building;  its 
foot  is  25^  ft.  from  the  building.  What  angle  does  it  make  with  the 
ground  ? 

(20.)  Two  buildings  are  on  opposite  sides  of  a  street  120  ft.  broad. 


26 


PLANE    TRIGONOMETRY 


The  height  of  the  first  is  55  ft. ;  the  angle  of  elevation  of  the  top  of 
the  second,  observed  from  the  edge  of  the  roof  of  the  first,  is  26°  37'. 
Find  the  height  of  the  second  building. 

(21.)  A  mark  on  a  flag-pole  is  known  to  be  53  ft.  7  in.  above  the 
ground.  This  mark  is  observed  from  a  certain  point,  and  its  angle 
of  elevation  is  found  to  be  25°  34'.  The  angle  of  elevation  of  the  top 
of  the  pole  is  then  measured,  and  found  to  be  34°  17'.  Find  the 
height  of  the  pole. 

(22.)  The  equal  sides  of-an  isosceles  triangle  are  each  7  in.  long ;  the 
base  is  9  in.  long.     Find  the  angles  of  the  triangle. 


Hint. — Draw  the  iierpendicular  BD.  BD  bisects  the  base,  and  also  the 
angle  A  BC. 

In  the  right  triangle  ABB,  AB—-]  in  ,  AD=\\  in.,  hence  ABD  can 
be  solved. 

Angle  C=angle  A,  angle  ABC— 2  angle  ABD 

(23.)  Given  the  equal  sides  of  an  isosceles  triangle  each  13.44  in., 
and  the  equal  angles  are  each  63°  21' 42";  find  the  remaining  parts 
and  the  area. 

(24.)  The  equal  sides  of  an  isosceles  triangle  are  each  377.22  in., 
the  angle  between  them  is  19°  55'  32".  Find  the  base  and  the  area 
of  the  triangle. 

(25.)  If  a  chord  of  a  circle  is  18  ft.  long,  and  it  subtends  at  the  centre 
an  angle  of  45°  31'  10",  find  the  radius  of  the  circle. 

(26.)  The  base  of  a  wedge  is  3.92  in.,  and  its  sides  are  each  13.25  in. 
long;  find  the  angle  at  its  vertex. 


THE   RIGHT    TRIANGLE  27 

(27.)  The  angle  between  the  legs  of  a  pair  of  dividers  is  64°  45',  the 
legs  are  5  in.  long;  find  the  distance  between  the  points. 

(28.)  A  field  is  in  the  form  of  an  isosceles  triangle,  the  base  of  the 
triangle  is  1793.2  ft. ;  the  angles  adjacent  to  the  base  are  each  53°  27' 
49".     Find  the  area  of  the  field. 

(29.)  A  house  has  a  gable  roof.  The  width  of  the  house  is  30  ft., 
the  height  to  the  eaves  25!  ft.,  the  height  to  the  ridge-pole  33^^  ft. 
Find  the  length  of  the  rafters  and  the  area  of  an  end  of  the  house. 

(30.)  The  length  of  one  side  of  a  regular  pentagon  is  29.25  in. ;  find 
the  radius,  the  apothein,  and  the  area  of  the  pentagon. 


Hint. — The  pentagon  is  divided  into  5  equal  isosceles  triangles  by  its  radii. 
Let  AOB  be  one  of  these  triangles.  A B=2g.2s,  in.;  angle  AOB=^  of 
36o°=72°.  Find,  by  the  methods  previously  given,  OA,  OD,  and  the  area 
of  the  triangle  A  OB. 

These  are  the  radius  of  the  pentagon,  the  apothem  of  the  pentagon,  and 
\  the  area  of  the  pentagon  respectively. 

(31.)  The  apothem  of  a  regular  dodecagon  is  2  ;  find  the  perimeter. 

(32.)  A  tower  is  octagonal ;  the  perimeter  of  the  octagon  is  153.7  ft. 
Find  the  area  of  the  base  of  the  tower. 

(33.)  A  fence  extends  about  a  field  which  is  in  the  form  of  a  regular 
polygon  of  7  sides;  the  radius  of  the  polygon  is  6283.4  ft.  Find  the 
length  of  the  fence. 

(34.)  The  length  of  a  side  of  a  regular  hexagon  inscribed  in  a  circle 
is  3.27  ft. ;  find  the  perimeter  of  a  regular  decagon  inscribed  in  the 
same  circle. 

(35.)  The  area  of  a  field  in  the  form  of  a  regular  polygon  of  9  sides 
is  483930  sq.  ft. ;  find  the  length  of  the  fence  about  it. 


28 


PLANE   TRIGONOME  TR  Y 


SOLUTION  OF   OBLIQUE  TRIANGLES  BY  THE  AID  OF 
RIGHT  TRIANGLES 

^,9.  Oblique  triangles  can  always  be  solved  by  the  aid  of 
right  triangles  without  the  use  of  special  formulas ;  the 
method  is  frequently,  however,  quite  awkward ;  hence,  in  a 
later  chapter,  formulas  are  deduced  which  render  the  solu- 
tion more  simple. 

The  following  exercises  illustrate  the  solution  by  means 
of  right  triangles : 

(I.)  In  an  oblique  triangle  a  =  3.72,  ^  =  47°  52',  C=i09"'  10';  find 
the  remaining  parts. 

The  given  parts  are  a  side  and  two  angles. 

C 


l^ 


'B 


^m/.— /f =[i8o°-(^+  C)\ 

Draw  the  perpendicular  CD. 

Solve  the  right  triangle  BCD. 

Having  thus  found  CD,  solve  the  right  triangle  A  CD. 

(2.)  In  an  oblique  triangle  a  =  89.7,  c=.  125.3,  B=  39°  8';  find  the 
remaining  parts. 

The  given  parts  are  two  sides  and  the  included  angle. 


THE  RIGHT   TRIANGLE 


29 


Hint. — Draw  the  perpendicular  CD. 

Solve  the  right  triangle  CBD. 

Having  thus  found  CD  and  AD{z=c  —  DB),  solve  the  right  triangle  ^ CZ>. 

(3.)  In  an  oblique  triangle  a  =  3.67, /J=  5.81,  ^  =  27°  23' ;  find  the 
remaining  parts. 

TAe  given  par  Is  are  two  sides  and  an  angle  opposite  one  of 
them. 


Either  of  the  triangles  ACB,  ACE'  contains  the  given  parts,  and 
is  a  solution. 

There  are  two  solutions  when  the  side  opposite  the  given  angle  is 
less  than  the  other  given  side  and  greater  than  the  perpendicular, 
CD,  from  the  extremity  of  that  side  to  the  base.* 

,  Hint. — Solve  the  right  triangle  A  CD. 

Having  thus  found  CD,  solve  the  right  triangle  CDB  (or  CDB'). 

(4.)  The  sides  of  an  oblique  triangle  are  «=  34.2,  ^  =  38.6,  £-  =  55.12; 
find  the  angles. 

The  given  parts  are  the  three  sides. 


C  =56.12 


*  A  discussion  of  this  case  is  contained  in  a  laicr  chapter  on  the  solution 
of  oblique  triangles. 


30 


PLANE    TRIGONOMETRY 


Hint.- 
Hence 


Let  DB=x, 

-x'^=crf=b'-{c-xf. 

a^=b'^  —  c^  +  2cx. 


In  each  of  the  right  triangles  ACD  and  BCD  the  hypotenuse  and  a  side 
are  now  known  ;  hence  these  trianglo  can  be  solved. 

(5.)  Two  trees,  A  and  B,  are  on  opposite  sides  of  a  pond.  The 
distance  of  A  from  a  point^C  is  297.6  ft.,  the  distance  of  B  from  C  is 
864.4  ft.,  the  angle  ACB  is  87°  43'  12".     Find  tlTe  distance  AB. 

(6.)  To  determine  the  distance  of  a  ship  A  from  a  point  B  on 
shore,  a  line,  BC,  ^00  ft.  long,  is  measured  on  shore  ;  the  angles,  ABC 
and  ACB,  are  found  to  be  67°  43' and  74°  21'  16"  respectively.  What 
is  the  distance  of  the  ship  from  the  point  B? 

(7.)  A  light-house  92  ft.  high  stands  on  top  of  a  hill;  the  distance 
from  its  ba.se  to  a  point  at  the  water's  edge  is  297.25  ft. ;  observed 
from  this  point  the  angle  of  elevation  of  the  top  is  46°  33'  15".  Find 
the  length  of  a  line  from  the  top  of  the  light-house  to  the  point. 

(8.)  The  sides  of  a  triangular  field  are  534  ft.,  679.47  ft.,  474.5  ft. 
What  are  the  angles  and  the  area  of  the  field  ? 

(9.)  A  certain  point  is  at  a  horizontal  distance  of  117^  ft.  from  a 
river,  and  is  11  ft.  above  the  river;  observed  from  this  point  the  angle 
of  depression  of  the  farther  bank  is  i  °  1 2'.  What  is  the  width  of  the  river? 

(10.)  In  a  quadrilateral /?^(7i9,^i?=  1.41,^(7=1.05,  CD=  1.76,  DA 
=  1.93,  angle  ^=75°  21^;  find  the  other  angles  of  the  quadrilateral. 


Jttu-Jy 


THE  RIGHT   TRIANGLE  31 


Hint. — Draw  the  diagonal  DB. 

In  the  triangle  ABD  two  sides  and  an  included  angle  are  given,  hence  the 
triangle  can  be  solved. 

The  solution  of  triangle  ABD  gives  DB. 

Having  found  DB,  there  are  three  sides  of  the  triangle  DBC  known,  hence 
the  triangle  can  be  solved. 

(II.)  In  a  quadrilateral  ABCD,  AB=i2.i,  AD  =  g.7,  angle  A  — 
^.7°  18',  angle  B  —  64=^  49',  angle  D=^  100°;  find  the  remaining  sides. 

Hi/ii.—Sohe  triangle  ABD  to  find  BD. 


CHAPTER   III 
TRIGONOMETRIC   ANALYSIS 

30,  In  this  chapter  we  shall  prove  the  following  funda- 
mental formulas,  and  shall  derive  other  important  formulas 
from  them : 

gin  {x  +  J/)  =  sin  a?  cos  y  +  cos  x  sin  j/,  (i  i) 

8ln(ac  — j/)  =  sinic  cosy —cos x  sin i/,  (12) 

cog(a7  + j/)  =  cosa;  cosy-slnx  siny,  (13) 

cos(iC- j/)  =  co8ac  cosy +  8lnic  siny;  (14) 

PROOF   OF   FORMULAS   (ll)-(l4) 

31.  Let  angle  ^6><2=.*^,  angle  QOP=y\  then  angle  AOP 

The  angles  x  and  y  are  each  acute  and  positive,  and  in  Fig.  i 
(:c-\-y)  is  less  than  90°,  in  Fig.  2  (x-^y)  is  greater  than  90°. 


In  both  figures  the  circle  is  a  unit  circle,  and  SP  is  perpendicular  to 
OA ;  hence  SP=. sin  {x  -\ry),  OS—  cos  (x  +  y). 


TRIGONOMETRIC  ANALYSIS  33 

Draw  DP  perpendicular  to  OQ  ; 
then  DP=%\v\y,     OD  =  cos)', 

angle  SPD  =  7ir\g\Q  A  OQ  =  x. 

(Their  sides  being  perpendicular.) 

Draw  DE  perpendicular  to  OA,  DH  perpendicular  to  SP. 
Si  n  (^  +  j)  =  5P=  ED  +  HP. 
ED  —  {?Ax\x)y.OD—'i\x\x  cosj. 

ED 

(For  OED  being  a  right  triangle,  -— -  =  sinx.) 

HP={cosx)  X  DP— COS  X  sin  J. 

fj  p 

(For  HPD  being  a  right  triangle,  — —  =  cos  jr.) 

Therefore,  gin(x  +  2/)  =  8lnic  cosy  +  cosx  giny.  (ll) 

Qos{x^y)  =  OS=OE-HD* 
OE = (cos  x)  X  OD  —  cos  .*•  cos  J. 

OE 
(For  OED  being  a  right  triangle, =  cosx.) 

HD  —  {s\nx)  X  Z^/'^sin;!:  sinj. 

fJ  D 

(For  PHD  being  a  right  triangle,  ---  =  sin  jc.) 

Therefore,  cos  (x  +  y)  =  cos  x  cos?/— sin  a?  sin?/.  (13) 

32,  The  preceding  formulas  have  been  proved  only  for 
the  case  when  x  and  y  are  each  acute  and  positive.  The 
proof  can,  however,  readily  be  extended  to  include  all  values 
of  X  and  y. 

Let  y  be  acute,  and  let  x  be  an  angle  in  the  second  quad- 
rant ;  then  jtr  =  (90°  +  ;r')  where  x'  is  acute, 
sin  {x ■\-y)  =  sin  (90°  +  x' -f- j) 

=  cos(;r'+j)  §24 

=  cos;ir'  cos^  — sin  x'  s\x\y 

=  sin  (90°  4-  x')  cosj  -f  cos  (90°  +  .«•')  sin  7      §  24 
=  sin.;tr  cosj-|-cos;ir  s\x\y. 

*  If  (x  +  1)  is  gre;iter  than  90°,  05  is  negative. 


34  ■     PLANE    TRIGONOMETRY 

Thus  the  formula  has  been  extended  to  the  case  where 
one  of  the  angles  is  obtuse  and  less  than  i8o°.  In  a 
similar  way  the  formula  for  cos(-r+^)  is  extended  to  this 
case. 

By  continuing  this  method  both  formulas  are  proved  to 
be  true  for  all  positive  values  of  x  and  y. 

Any  negative  angle  y  is  equal  to  a  positive  angle  y' ,  minus 
some  multiple  of  360°.  The  functions  of  y  are  equal  to 
those  of  y' ,  and  the  functions  of  {xi-y)  are  equal  to  those 
of  {x+y').  §  9 

Therefore,  the  formulas  being  true  for  (x  -{-y'),  are  true  for 
(x+y). 

A  repetition  of  this  reasoning  shows  that  the  formulas  are 
true  when  both  angles,  x  and  y,  are  negative. 

33.  Substituting  the  angle  —y  for  y  in  formula  (u),  it 
becomes 

sin{x—y)  =  sln  x  cos(— _;^)4-cos;ir  sin  {—y). 
But  cos(  — j/)  =  cosj/,  and  sin  (— j)=  — sin  J.      §23 

Therefore,  8in(ic  —  y)  =  sinx  cony  —  cos x  siny.  (12) 

Substituting  {—y)  for y  in  formula  (13),  it  becomes 
cos{x—y)  =  cosx  cos(— J/)— sin;jr  sin{—y), 
=:Cos;r  cosj)/  +  sin;t- sin_y. 
Therefore,  co»{x - y)  =  co» x  cony  +  niux  slii?/.*  (14) 

EXERCISES 

34.  (r.)  Prove  geometrically  where  x  and  ^  are  acute  and  positive : 

sin(^— ^)  =  sin  jr  cosjy  —  cos^  sin^, 
cos(;ir — j)  =  cosjr  cos/ -f- sin  x  sin^. 

*  Formulas  (12)  and  (14)  are  proved  geometrically  in  §  34.  The  geometric 
proof  is  complicated  by  the  fact  that  OD  and  DP  are  functions  of  —y,  while 
the  functions  of  y  are  what  we  use. 


TRIGONOMETRIC  ANAL  YSIS 


35 


Hiiit.—Kw^t.  AOQ-x,  angle  POQ=y,  and  angle  AOP—(x-y). 

Draw  PD  perpendicular  to  OQ. 
Then /?/'= sin (—j')= —sin  v;  but  DP  is  negative,  therefore  PD  taken 
as  positive  is  equal  to  sin  y : 

OD=cos{  —y)  =  cos  y. 

Angle  HPD=a.ng\e  AOQ=x.  their  sides  being  perpendicular. 

Draw  DH  perpendicular  <.o  SP,  DE  perpendicular  to  OA. 

%xn{x-y)=SP=ED-PH. 

P'rom  right  triangle  OED,    ED.—  {^mx)y.OD=i%\nxcosy. 

From  right  triangle  DHP,    P//=(cosx)  X  PD=cosx  sin  j. 

Therefore,  sin  (;r— j)=sin  x  cos^  — cos  jc  sinjj'. 

Cos  (;r  -;)  =  C>  .S  '=  OE  +  D//. 

From  right  triangle  OED,      OE={cosx)xOD=cosx  cosy. 

From  right  triangle  D/IP,     D//={smx)x  PD  =  s'mx  sinj. 

Therefore,  cos(jr— j)=cos jr  cosj+sin jr  sinj. 

(2.)  Find  the  sine  and  cosine  of  (45°+;r),  (30°— jr),  (60°+. r),  in  terms 
of  sin  X  and  cos  .r. 

(3.)  Given  sin;r=|,  sin/  =  ^,  x  and  y  acute;  find  sin(jr+j)  and 
sin  (.r — J'). 

(4.)  Find  the  sine  and  cosine  of  75°  from  the  functions  of  30°  and  45°. 
J/inL—  75°=(45°  +  30°). 

(5.)  Find  the  sine  and  cosine  of  1 5°  from  the  functions  of  30°  and  45°. 

(6.)  Given  x  and  j,  each  in  the  second  quadrant,  sin  x  =  ^,  sin_y  =  J ; 
find  s\n{x-\-y)  and  cos(jr— ^). 

(7.)  By  means  of  the  above  formulas  express  the  sine  and  cosine  of 
(i8o°  — ;r),  (i8o°+;jr),  (270°— .r),  (270°+^),  in  terms  of  sin^  and  cos  or. 

(8.)  Prove  sin  (60°+ 45")  + cos  (60° +  45°)  =  cos  45°. 

(9.)  Given  sin45°  =  ^\/2,  cos45°=^ -v/2  ;  ^^^  sin  90°  and  COS900. 

(10.)  Prove  that  sin(6o°-|- ;r)  —  sin  (60°  —  .r)  =  sin-i-. 


36  PLANE   TRIGONOMETRY 

TANGENT  OF  THE   SUM  AND   DIFFERENCE  OF  TWO  ANGLES 

cos (-1:4- j)     cos;r  cos^— sin;ir  sinj/ 
Dividing  each  term  of  both  numerator  and  denominator 
of  the  right-hand  side  of  this  equation  by  cos;r  cosj,  and 

remembering  that  —  =  tan,  we  have 
^  cos 

,     .     ,       tan  X,  +  tan  y  ,     . 

*""(^  +  ^>  =  l-tan;«tany-  OS) 

In  a  similar  way,  dividing  formula  (12)  by  formula  (14),  we 

obtain 

,  ,       tan  05  -  tan  y  .  ^. 

tan  (x  -  w)  =  =— — T — ^-.  (16) 

^        ^'     l+tana^lan;/  V     / 

FUNCTIONS   OF   TWICE   AN   ANGLE 

3(i,  An  important  special  case  of  formulas  (11),  (13),  and 

(15)  iswhen  j  =  ;ir;  we  then  obtain  the  functions  of  2x  in 

terms  of  the  functions  of  x. 

From  (i  i),  sin(;tr-+-;ir)=:isin;ir  cos;tr-f-cos;r  sin;r. 
Hence  sin  2ic  =  2  sin  a?  cos  a?.  (17) 

From  (13),       cos2ic  =  cos''ic  —  sln*ic.  (18) 

Since        cos*;ir=:  l  — sinV,  and  sin'';r=  l  — cos";}:, 

we  derive  from  equation  (18), 

cos  2  A'  =  I  —  2  sinV,  (19) 

and  cos2;ir=2  cosV— I.  (20) 

From  (15),       tan2x  =  ^^l^^.  (21) 

FUNCTIONS  OF  HALF  AN  ANGLE 

,?7.  Equations  (19)  and  (20)  are  true  for  any  angle;  there- 
fore for  the  angle  \x. 

From(i9),  cos;r=i  — 2  sin'^;r; 


TRIGONOMETRIC  ANALYSIS  37 

I  — zo%x 


or  sin''^;i-  = 

1         r  -     1         _L     /l  -COS a?  ,^  V 

therefore,  sm^x=±\/ —  (22) 

From  (20),  cos;r  =  2  o.O's^x  —  l  ; 

„,         14-  cos;ir 
or  cos  \x  = ; 

2 

1        r                               1                /'  +  cosa?  .     V 

therefore,  cos-|aj=:rfc\/ — -•  (23) 

Dividing  (22)  by  (23),  we  obtain 

1  ,      /*  ~  cos  a?  ,     X 

FORMULAS   FOR   SUMS  AND    DIFFERENCES   OF   FUNCTIONS 

3S»  Fronn  formulas  (ii)-(i4),  we  obtain 

sin  {x  •\- y)-\-'i\n  (;ir  — j)  =  2sin;tr  C0S7  ; 
sin  (-r  +  jj/)  — sin  {x  —  y)=.2zo%x  sinj ; 
cos  {x  ■\-y)  +  cos  {x—j')  =  2  cos;r  cosj/ ; 
cos  (;i: +_)/)  — cos  (;f—jj/)=  —  2sin;tr  sinj. 
Let  u  =  {x+y)  and  v—X^—y); 

then  x  =  ^{u  +  7'),  j/  =  ^{u  —  v). 

Substituting  in  the  above  equations,  we  obtain 

sin ««  +  sin «' =  2  tiin^{u-\-v)cos^{ti  —  v)',  (25) 

sin'«-sint'  =  2cos-|(i«  +  r)8in^(t*- v);  (26) 

cosM  +  cosr=2co8-|(»«+r)cos|^(?e  — r);  {zy) 

cos?t-cost7=-2sin^(w  +  ^0  8in^(u-r).         (28) 
Dividing  (25)  by  (26), 

8in?*  +  sint»     tan^{u+v)  ^ 


sin  u— sin  V     tan^{ti—v) 


(29) 


EXERCISES 
39.  Express  in  terms  of  functions  of  x,  by  means  of  the  formulas 
of  this  chapter, 

45071 


38  PLANE   TRIGONOMETRY 

(I.)  Tan(i8o°  — jr);  tan(i8o°4-^). 
(2.)  The  functions  of  {x  —  i8o°). 
(3.)  Sin(jir  — 90°)  and  005(^—90°). 
(4.)  Sin  (or  —  270°),  and  cos  {x  —  270°). 
(5.)  The  sine  and  cosine  of  (45°— a:);  of  (45°-|-jc). 
""    (6.)  Given  tan  45°=  i,  tan  ■},d^  —  ^  y/y,  find  tan  75°;  tan  15°. 

(7.)  Provecot  (ic  +  M)  = ~^ — .  (30) 

Hint. — Divide  formula  (13)  by  formula  (11). 

(8.)  Prove  cot,x-w)  = .  (31) 

(9.)  Prove  cos  (30 -|-/)  —  cos  (30°  —y)  =  —  sin_y. 
(10,)  Prove  sin  3;*:  =  3  sin  X  —  4  sin';r. 

Hint. — Sin  3x  =  sin  (jr+2Jr). 
(II.)  Prove  COS  3Jr  =z  4  cos'  —  3  cos^r. 
(12.)  If  X  and  y  are  acute   and   tan;jr  =  J,  tanj  =  J,  prove  that 

(j'+j)  =  45°- 

^  1-.  ,  /     .       ^v      1  "|-  tan  X 

(13.)  Prove  that  tan(;r-f  45°)  =  — . 

I  —  tan  X 

(14.)  Given  sinj  =  |  andj  acute;  find  sin|j,  cos^/,  and  tan^/. 

(15.)  Given   cos;«r=  — |  and  x  in  quadrant  II;    find    sin  2-r  and 
cos  2.r. 

(16.)  Given  cos45°  =  |  y/z  \  find  the  functions  of  22^°. 

(17.)  Given  tan  jr  =  2  and  x  acute  ;  find  tan  \x. 

(18.)  Given  cos30°  =  ^y'3;  find  the  functions  of  15°. 

(19.)  Given  cos  90°  =  ©;  find  the  functions  of  45°. 

(20.)  Find  sin^jr  in  terms  of  sin;r. 

(21.)  Find  cos5,r  in  terms  of  cos  j-. 

(22.)  Prove  sin  {x  -\-y  ■\-z)  =  sin  x  co%y  cos  ^'^cos  x  sin  v  cos  S'+cos  x 
cos/  sins-  —  sin  jr  sin/  sin 2'. 

///«/■.— Sin  (jr+/  +  3)=sin(jr+j')  cos 3  +  cos ( jf  +  j)  sin  2. 

(23.)  Given  tan  2jr  =  3  tan  x ;  find  x. 

(24.)  Prove  sin  32° -|- sin  28°  =  cos  2". 

(25.)  Prove  tan  x ■\-  cot  jr  =  2  esc  ix. 

(26.)  Prove  (sin^j'  +  cos^;r)'r=  i  +  sin  jr. 

(27.)  Prove  (sin  \x  —  cos  ^.r)'^  =  i  — sin  x. 


TRIGONOMETRIC  ANALYSIS  39 

(28.)  Prove  cos  ^x  —  cos^r  —  sin*.r. 

(29.)  Prove  tan  (45°  -\-x)-\-  tan  (45^  —  ^)  =  2  sec  2x. 

■               2  tan  JT 
(30.)  Prove  sin2x  =  — . — . 

,  ^  I  —  tan^r 

(31.)  Prove  cos2.r  = ■— . 

■^  1  +  tan^.r 

,   „  i-fsin2.r      /tan.t-t-iV 

(32.;  Prove : —    )  • 

I — sin  2^       \tanx— 1/ 

^  r.  ,  sin.r 

(33.)  Prove  tan^ji: 


(34.)  Prove  cot^.r 


I  +COS  JT 

sin.r 


I  —  COS  X 

cos  X  —  cosy 


(35.)  Express  as  a  product 

COS  X  +  cos/ 

.^.    ,  cosjf  — COST      —2  sini(jr  +  r)  sini{x—y) 

Hint. —  =- -^ '- 1— — — — 

cos x  + cos/         2  cos^(j:  +  v)  cos^(jr— j) 

=  —  tan |{ jf  +j')  tan \ {x  —y). 
tan  X  +  tan  J/ 


(36.)  Express  as  a  product 
(37.)  Prove  1 — tan  Ji- tan  J 


cot  X  -f  cot_j/ 
cos  (x  -\-y) 
cos-r  cosj' 


THE   INVERSE   TRIGONOMETRIC   FUNCTIONS 

4:0,  Dcf. — The  expressions  sin-'^,  cos— '^,  tan-'^;,  etc.,  de- 
note respectively  an  angle  whose  sine  is  a,  an  angle  whose 
cosine  is  a,  an  angle  whose  tangent  is  a^  etc.  They  are 
called  the  inverse  sine  of  a,  the  inverse  cosine  of  a,  the 
inverse  tangent  of  a,  etc.,  and  are  the  inverse  trigono- 
metric functions. 

Sin— 'rt  is  an  angle  whose  sine  is  equal  to  a,  and  hence  de- 
notes, not  a  single  definite  angle,  but  each  and  every  angle 
whose  sine  is  a. 

*  Since  quantities  cannot  be  added  or  subtracted  by  the  ordinary  operations 
with  logarithms,  an  expression  must  be  reduced  to  a  form  in  wliich  no  addition 
or  subtraction  is  required,  to  be  convenient  for  logarithmic  computation. 


40  PLANE    TRIGONOMETRY 

Thus,  if  sinjr=^,  jr=3o°,  150°,  (30°  +  36o°),  etc., 

and  sin- '^=30°,  150°,  (30°  +  36o°),  etc. 

Remark. — The  sine  or  cosine  of  an  angle  cannot  be  less  than  —  i 
or  greater  than  +  i;  hence  sin-'«  and  cos-'a  have  no  meaning  unless 
a  is  between  — i  and  -j-  i.  In  a  similar  manner  we  see  that  sec-V? 
and  csc-'a  have  no  meaning  if  a  is  between  —  i  and  •\- 1. 

EXERCISES 

4:1,  (I.)  Find  the  following  angles  in  degrees: 

s\n-^\yji,  tan-'(— i),  sin-'( — W 

COS~'^,  COS"' I, 

(2.)  If  ;r  =  cot-4.  find  tan;r. 

(3.)  If  X  =  sin-'f,  find  cos^ir  and  tan  x. 

(4.)  Find  sin  (tan— ^  Vi)- 
(5.)  Find  sin  (cos— I). 
(6.)  Find  cot  (tan— xV)- 

(7.)  Given  sin-'<3:  =  2  cos— «,  and  both  angles  acute;  find  a. 
(8.)  Given  sin-'a  =  cos— «: ;  find  the  values  of  sin— a  less  than  360°. 
(9.)  Given  tan-'i  =| tan-'o,  and  both  angles  less  than  360°;   find 
the  angles. 

(10.)  Given  sin— rt  =  cos— rt  and  sin— ^-f- cos— rt!  =450°;  find  sin-'rt. 

(II.)  Prove  sin  (cos-'rt)=:±  -y/ \  — a^. 

Hint. —  Let  jr=cos-'rt ;  then  a  =  c6sx, 

sin  x=  ±  y  I  —  cos^'j:  =  ±  y  i  — «'. 

(12.)  Prove  tan(tan— rt:+tan-'^)=    _  ,/' 

^  ,        (I  —  b 

( 1 3.)  Prove  tan  (tan-'rt  —  tan-'^)  =  ■         ,  • 

(14.)  Prove  cos  (2  cos-'^)  =  2a^  —  i . 

(15.)  Prove  sin(2cos— «)=:db2rt  v^i  —  a'. 

2a 
(16.)  Prove  tan (2  tan-' rt)= -• 

(17.)  Provecos(2tan-'a)= 


i+a" 
(18.)  Prove  sin (sin-'rt  +  cos— ^)  —  ab±. "/(i  —«»)(!  — **). 


CHAPTER   IV 

THE    OBLIQUE    TRIANGLE 

DERIVATION   OF   FORMULAS 

4:2,  The  formulas  derived  in  this  and  the  succeeding 
articles  reduce  the  solution  of  the  oblique  triangle  to  its 
simplest  form. 

C  C 


HG.  3 


Draw  the  perpendicular  CD.     Let  CD^^^Ii, 

h      .      . 
Then  -  =  sm^; 

o 

(In  Fig.  2    -'=sin(i8oO-^)  =  sin^) 

and  -  =  sin  B. 

a 

(In  Fig.  3  -=sin(i8o°-i9)  =  sin^.) 
a 

By  division  we  obtain, 


sin^ 


/U^         />ivV  (\ 


^    (32)      ^ 


h     sin  B 

Remark. — This  formula  expresses  the  fact  that  the  ratio  of  two  sides  of  an 
oblique  triangle  is  equal  to  the  ratio  of  the  sines  of  the  angles  opposite,  and 
does  not  in  any  respect  depend  upon  which  side  has  been  taken  as  the  base. 
Hence  if  the  letters  are  advanced  one  step,  as  shown  in  the  figure,  we  obtain, 
as  another  form  of  the  same  formula, 


42 


PLANE   TRIGONOMETRY 


b  _  sin  B 

c      sinC 
Repeating  the  process*  we  obtain 

c  _  sin  C 

rt      sin/i 

The  same  procedure  may  be  applied  to  all  the  formulas  for  the  solution  of 
oblique  triangles.      Henceforth  only  one  expression  of  each  formula  will  be  given. 

Formula  (32)  is  used  for  the  solution  of  triangles  in  which 
a  side  and  two  angles,  or  two  sides  and  an  angle,  opposite  one 
of  them  are  given. 

4:3,  We  obtain  from  formula  (32)  by  division  and  compo- 
sition, a—b%\\\A  —  sA\\B 
a-vb~  s\\\A-\-  sin  B ' 
By  formula  (29),  denoting  the  angles  by  A   and  B,  in- 
stead of  u  and  v, 

%\rvA-^mB    \.d.^\{A—B) 


Therefore, 


sin^-f-sin^     \2.\\\{A-vB) 
a—b     tan  1(4— JB) 


(33) 


a-l-6     tAn^{A  +  B) 
This  formula  is  used  for  the  solution  of  triangles  in  ivhich 
tivo  sides  and  the  included  angle  are  given. 
44.  Whether  A  is  acute  or  obtuse,  we  have 
C  C 


(If  A  is  acute  (Fig.  i),  AD  —  b  cos  A,  DB  =  AB  -  AD  — c  -  b  coi,  A,  CD— 
bixnA.  UAh  obtuse  (Fig.  2),  ^Z>  =  3cos  (i8o°-^)  =  -  bco%A,  DB=AB 
+  AD=c  —  bcosA,  CD  =  b  sin(i8o°- /i)=  <J  sin^.) 


THE   OBLIQUE    TRIANGLE  43 

a^  =  {c-b  cosAy  +  {d  slnAf, 
=  f'-2  dc  cosAi-d'  (cosM  +sinM). 

Therefore,  a^—b^-\^c^-2bc  cos  JL.  (34) 

TJiis  foriiiula  is  2iscd  in  deriving  formula  (37). 

//  is  also  used  in  the  solution  without  logarithms  of  tri- 
angles of  which  tzvo  sides  and  the  included  angle  or  three 
sides  are  given. 

7,2     I         2  2 

•io.   From  formula  (34),  cosy^  = 7 

From  formula  (22),  §  37, 

b'^-c'-a' 


2  sxn^^A  =  I  — cos^  =  I  — 
Hence  2  sin'^^r^ 


2bc-\rd'-b''-c^ 


2bc 

__a'-(b-cy 
~        2bc 

_{a  —  b  +  c){a-\-b—c) 
~  2bc 

Let  i'  = ,  then  {a—b-k-c^  —  2{s  —  b),  and  {a-\-b  —  c^  = 

2{s—c). 

2  {s—b){s—c) 


Substituting,  2  sin^i-^  = 


be 


Hence      '  sin^^  =./(ir:Mz^).''  "         (35) 

V  be 

From  formula  (23),  §  37, 

M   /;      .  ,  A      2bc  +  b'  +  e'-a'' 

2  cos^A  =  I  +cosyi  = 7 , 

^  2  be 

_2s{s'  a) 
"^       b~e 

*  111  extracting  the  root  the  phis  sign  is  cliosen  because  it  is  known  that 
sin^/4  is  positive. 


44 


PLANE    TRIGONOMETRY 


Hence  zo^\A  =J'^tZ^). 

V      be 

Dividing  (35)  by  (36),  we  obtain 


{s—d){s  —  b){s—c) 


Let 


){s-b){s-c) 


tan  \  A 


s — a 


(36) 

(37) 


(38) 


Formulas  (37)  and  (38)  are  used  to  fi?td  the  angles  of  a  tri- 
angle when  the  three  sides  are  given. 


FORMULAS   FOR   THE   AREA   OF   A   TRIANGLE 

46*.  Denote  the  area  by  5. 


(In  Fig.  I,  CD=asinB;  in  Fig.  2,  CZ>=r  asin(i8o°-^)  =  asin^. 

In  Figs.  I  and  2,         S=i^c.CD. 

Hence  S=^acsinB, 

From  formula  (17), 


(39) 


THE   OBLIQUE   TRIANGLE  45 

Substituting  for  sin^^  and  zo-s^B  the  values  found   in 
formulas  (35)  and  (36),  we  obtain 

s\ryB=^—\s{s  —  a){s  —  b){s  —  c). 


Therefore, 


S = Kj  s{s  —a)(s— b)(s — c). 

This  formula  may  also  be  written, 
S=sK. 

Fonnida  (39)  is  used  to  find  the  area  of  a  triafigle  when 
two  sides  and  the  included  angle  are  known;  formula  (40)  or 
formula  (41),  when  the  three  sides  are  known. 


(40) 
(41) 


THE   AMBIGUOUS    CASE 
47.  The  given  parts  are  two  sides,  and  the  angle  opposite 
one  of  them. 

Let  these  parts  be  denoted  by  a,  b,  A. 


B  B  \     A 


If  a  is  less  than  b  and  greater  than  the  perpendicular  CD 
(Fig.  i),  there  are  the  two  triangles  ^4 6"^  and  ACB',  whicli 
contain  the  given  parts,  or,  in  other  words,  there  are  two 
solutions. 

If  a  is  greater  than  b  (Fig.  2),  there  is  one  solution. 

If  a  is  equal  to  the  perpendicular  CD,  there  is  one  solu- 
tion, the  right  triangle  ACD. 


46 


PLANE    TRIGONOMETRY 


If  the  given  value  of  a  is  less  than  CD,  evidently  there 
can  be  no  triangle  containing  the  given  parts. 

Since  CZ>=^  sin /4,  there  is  no  solution  when  «  <  b'^xwA  ;  there  is  one 
solution,  the  right  triangle  ACD  when  a=.b%\x).A\  there  are  two  solutions 
when  a<.h  and  >  h  sin  ^. 

48.  Case  I. — Given  a  side  and  two  angles. 

EXAMPLE 
Given  a  =  36.738,  A  =  36°  55'  54".  B  =  72°  5'  56". 

C=i8oO— (v4  +  ^)=i8o°— 109°  I'  5o"=:7o°  58'  10". 


To  find  b. 

To  find  c. 

b      sin  5 

c      sin  C 

a      %\nA 

a      sin  A 

log  rt  =  i. 56512 

loga=r. 56512 

log  sin  ^=9.97845  — 10 

logsinC=9.97559-io 

colog  sin  A  =0. 22 1 23 

colog  sin  A  =0. 22123 

log /^=  1. 76480 

log  r=  1. 76194 

<J=58.i84 

r=57.8o 

Check. 

ermine  h  from  c,  C,  and  B  by  the  formula 

b-a_i9iX\\{B-A) 

b-Va      tani(^+^)" 
This  check  is  long,  but  is  quite  certam  to  reveal  an  error.     A  check  which  is 
shorter,  but  less  sure,  is 

b  _  sin  B 

c     sin  C 

Solve  the  following  triangles  : 
(I .)  Given  a  =  567.25,  A  =  11°  is',  B  =  47°  12'. 
(2.)  Given  a  =  783.29,  A  =  8i°  52',  B  =  42°  27'. 
(3.)  Given  c=  1 125.2,  A  =  79°  is',  B=  55°  il', 
(4.)  Given  <5=:  15.346,  ^=15°  51',  C=58°  10'. 
(5.)  Given  rt  =  5301.5,  ^=69°  44',  C=4i°  18'. 
(6.)  Given  ^=1002.1,  A  =  48°  59',  €  =  76°  s'- 

40.  Case  II. — Given  two  sides  of  a  triangle  and  the  angle 
opposite  one  of  them. 


THE   OBLIQUE    TRIANGLE 


47 


EXAMPLE 
Given  a  =  23.203,  (5  =  35.121,  A  =  36°  8'  10". 

C 


To  find  B  and  B' . 
sin  B     b 
%\Vi.A     u 
log  ^=1.5^556 
log  sin/^=g.  77064  — 10 
cologrt=8. 63445  — 10 
log  sin  i9=9. 95065  — 10 
5=63°  12' 
^'=:i8o°-.(9=ii60  48' 

To  find  C  and  C' . 
C  =i8o°-(^  +^)=8o°  39'  50" 
C"  =  i8o°-(^  +  5')=27°3-  50" 


I'o  find  c  and  c  . 
c     sin  C 
a~%mA 
log  a=i. 36555 
log  sin  C=g.9942i  — 10 
colog  sin  ^=0.22936 
log  f=i. 58912 
<r=38.825 

log  rt  =  i. 36555 
log  sin C"  =9. 65800— 10 
colog  sin /^  =0.22936 
log  f'  =  i. 25291 
^'  =  17.902 


Check. 

Determine  b  from  c,  C,  and  B  by  the  formula 

^-^_tan|.(.g-^) 

b+a~ta.n\{B  +  A)' 

This  check  is  long,  but  is  quite  certain  to  reveal  an  error.     A  check  which  is 

shorter,  but  less  sure,  is 

^_sinj9 
c     sin  C 


(I.)  How  many  solutions  are  there  in  each  of  the  following? 
(I.)  ^  =  30°,  a  =  is,  b  —  2o; 
(2.)  A  =  30°  a  =  io,b  =  20\ 
y  (3.)  ^  =  300  a  =  8,  ^  =  20; 

(4.)  ^  =  37023',  a  =  9. 1,  ^  =  7. 5. 


> 


48 


PLANE    TRIGONOMETRY 


Solve  the  following  triangles,  finding  all  possible  solutions: 

(2.)  Given  A  =  147°  12',  «  =  0.63735.  (^  =  0.34312. 

(3.)  Given  ^=   24°  31',  a  =  1.7424,    <^  =  o.96245. 

(4.)  Given  ^  =   21°  21',  «  =  45^93,    d=    56.723. 

(5.)  Given  ^  =    61°  16',  a  =  9.5124,    6=    12.752. 

(6.)  Given  C=   22°  32',  «  =0.78727,  <:  =  0.4731 1. 

SO,  Case  III. — Given  two  sides  and  the  included  angle. 


EXAMPLE 

Given  ^  =  41.003,  ^  =  48.718,  C  =  68o  33'  58";   find  the  remaining 
parts  and  the  area. 


To  find  A  and  B. 

iax\\{B  —  A)  _  b  —  a 
tani(/5  +  ^)"~"^  +  «" 

b-a=    7.715 

b  +  a  =  89.721 

i(^  +  ^)  =  55°43'i". 

log  (^  —  «)  =  0.88  734 
colog  {b  +  a)  =  8.04710  —  10 
log  tan \{B  +  A)=zo.  16639 
log  tani(,5  —  ^)  =  9.10083  —  10 
\{B  -A)=    7°  11'  20" 
i(^  +  ^)  =  55°43'    I" 


B  =  62°  54'  21" 
y4=48°3i'  41" 


To  find  c. 
£  _  sin  C 
a      sin  .,4 
logrt  =  1.61281 
log  sin  C=  9.96888  -10 
colog  sin  A  z=.o.  12535 
log  f  =  1 .  70704 
<:=    50.938 

To  find  the  area. 
Sz=-\ab  <a\nC 
logi  =  9-69897—10 
logrt  =  1.61281 
log<^=  1.68769 
log  sin  C=  g. 96888  — 10 
log  A"  =2.96835 
5=    929.72 


Check, 
sin  C       c 
sin  B       b 
log  sin  B  =  9.94951  —  10 
log  c  -=1. 70704 
colog  b  —  8.31231  —  10 
log  sin  C  —  9.96886  —  ID 


THE  OBLIQUE   TRIANGLE 


49 


Solve  the  following  triangles,  and  also  find  their  areas 
(I.)  Given  y^=  41°  15',  <^=o.i4726,  c=o.io97i. 


)  Given  C=  58°  47',  (5=11.726,  a=\6.\\T. 
)  Given  Bz=.  49°  50',  rt!  =  i03  74,  <r=:99.975. 
)  Given  A=  33°  31',  (^=0.32041,  ^=0.9203. 
)  Given   C=I28°   7,  <5=  17.738,    a—bo.^'ji. 


ol.  Case  IV. — Given  the  three  sides. 

EXAMPLE 
Given  a  =  32.456,  <5  =  41.724,  c=  53.987  ;  find  the  angles  and  area. 
J  =  64.084 


{s  —rt)r=  31.628 

{s  —  b)  =  22.360 
{s  —  c)  —  10.097 


,._.  /(: 


v^^ 


a){s  -  />){s  -  c) 


log  (j  —  ^r)  =  1.50007 
log  (j -<!')=  1. 34947 
log  (j  —  r)=l. 00419 
colog  ^  =  8.19325  — 10 

2)2.046q8 
log  A'=  1.02349 

To  find  A. 

K 

tan+//  = • 

s  —  a 

log  ir=  1.02349 

log  (j—rt) =1.50007 


sub. 


To  find  B. 

\.z.Vi\B— -.  • 

log  A'=  1. 02349 
log  (j -(!')  =  1. 34947 


sub. 


log  tan  I  ^=9.67402— 10 
i^=25°  16'  16" 
B=io°  32'  32" 


To  find  C* 

K 

taniC= 

^         s—c 

log  A'=  1.02349 

log  (j'— f)=i.oo4i9 


sub. 


log  tan^C=o.oi930 

^C=46°  16'  22" 
C=92°  32'  44" 


log  tan|^  =9. 52342  —  10 
i^  =  i8°  27'  23" 
^=36°  54'  46" 

Check. 
(^+i9+Cy=i8o°o'  2", 

Find  the  angles  and  areas  of  the  following  triangles : 
(I.)  Given  ^=38.516,  (^=44.873,  ^=14.517. 
(2.)  Given  rt!  =  2.ii58,  1^=3.5854,  r=3.5679. 

.    *  C  could  be  found  from  {A  +.5)=(i8o°—  C),  but  for  the  sake  of  the  check  it 
is  worked  out  independently. 
4 


50  PLANE    TRIGONOMETRY 

(3.)  Given  ^  =  82.818,  ^=99.871,  ^=36.363. 
(4.)  Given  a—^b.^Zf),  ^=11.698,  ^=33.328. 
(5.)  Given  0  =  113.08,  ^=131.17,  f  =114.29. 
(6.)  Given  a  =  .9763,^=1.2489,^=1.6543. 

EXERCISES 

52.  (1.)  A  tree, /^,  is  observed  from  two  points,  B  and  C  1863  ft. 
apart  on  a  straight  road.  The  angle  BCA  is  36'' 43',  and  the  angle 
CBA  is  57°  21'.  Find  the  distance  of  the  tree  from  the  nearer 
point. 

(2.)  Two  houses,  A  and  B,  are  3876  yards  apart.  How  far  is  a  third 
house,  C,  from  A,  if  the  angles  ABC  and  BAC  d^r^  49°  17'  and  58°  18' 
resf)ectively  } 

(3.)  A  triangular  lot  has  one  side  285.4  ft.  long.  The  angles  adja- 
cent to  this  side  are  41°  22'  and  31°  19'.  Find  the  length  of  a  fence 
around  it,  and  its  area. 

(4.)  The  two  diagonals  of  a  parallelogram  are  8  and  10,  and  the 
angle  between  them  is  53°  8' ;  find  the  sides  of  the  parallelogram. 

(5.)  Two  mountains,  A  and  B,  are  9  and  13  miles  from  a  town,  C \ 
the  angle  ACB  is  71°  36'  37".  Find  the  distance  between  the  moun- 
tains. 

(6.)  Two  buoys  are  2789  ft.  apart,  and  a  boat  is  4325  ft.  from  the 
nearer  buoy.  The  angle  between  the  lines  from  the  buoys  to  the 
boat  is  16°  13'.  How  far  is  the  boat  from  the  farther  buoy.'  Are 
there  two  solutions? 

(7.)  Given  <z  =164.256,  r  =  19.278,  C=i6°  19'  11";  find  the  differ- 
ence in  the  areas  of  the  two  triangles  which  have  these  parts. 

(8.)  A  prop  13  ft.  long  is  placed  6  ft.  from  the  base  of  an  embank- 
ment, and  reaches  8  ft.  up  its  face;  find  tjje  slope  of  the  embank- 
ment. 

(9.)  The  bounding  lines  of  a  township  form  a  triangle  of  which  the 
sides  are  8.943  miles,  7.2415  miles,  and  10.817  miles;  find  the  area 
of  the  township. 

(10.)  Prove  that  the  diameter  of  a  circle  circumscribed  about  a 
triangle  is  equal  to  any  side  of  the  triangle  divided  by  the  sine  of  the 
angle  opposite. 


THE  OBLIQUE    TRIANGLE 


SI 


Hint. — By  Geometry,       angle/i6>iff=2C 

Draw  OD  perpendicular  to  AB. 

Angle  DOB-\AOB-C. 

DB—r  sin  DOB—r  sin  C. 

Hence  ^=2rsinC, 

c 
or  21-=-^—. 

sin  6 


(II.)  The  distances  AB,  BC,  and  AC,  between  three  cities,  A,  B, 
and  Care  12  miles,  14  miles,  and  17  miles  respectively.  Straight  rail- 
roads run  from  A  to  B  and  C.     What  angle  do  they  make  ? 

(12.)  A  balloon  is  directly  over  a  straight  road,  and  between  two 
points  on  the  road  from  which  it  is  observed.  The  points  are  15847 
ft.  apart,  and  the  angles  of  elevation  are  found  to  be  49°  12'  and 
53°  29'  respectively.  Find  the  distance  of  the  balloon  from  each  of 
the  points. 

(13.)  To  find  the  distance  from  a  point  ^  to  a  point  B  on  the  op- 
posite side  of  a  river,  a  line,  AC,  and  the  angles  CAB  and  ACB  were 
measured  and  found  to  be  315.32  ft.,  58^  43',  and  57°  13'  respectively. 
Find  the  distance  AB. 

(14.)  A  building  50  ft.  high  is  situated  on  the  slope  of  a  hill.  From 
a  point  200  ft.  away  the  building  subtends  an  angle  of  12°  13'.  Find 
the  distance  from  this  point  to  the  top  of  the  building. 

(15.)  Prove  that  the  area  of  a  quadrilateral  is  equal  to  one-half 
the  product  of  the  diagonals  by  the  sine  of  the  angle  between 
them. 

(16.)  From  points  A  and  B,  at  the  bow  and  stern  of  a  ship  respec- 
tively, the  foremast,  C,  of  another  ship  is  observed.  The  points  A 
and  B  are  300  ft.  apart ;  the  angles  ABC  and  BAC  are  found  to  be 


52  PLANE   TRIGONOMETRY 

65°  31' and  1 10°  46'  respectively.     What  is  the  distance  between  the 
points  A  and  C  of  the  two  ships  ? 

(17.)  Tavo  steamers  leave  the  same  port  at  the  same  time  ;  one  sails, 
directly  northwest,  12  miles  an  hour;  the  other  17  miles  an  hour,  in 
a  direction  67°  south  of  west.  How  far  apart  will  they  be  at  the  end 
of  three  hours  } 

(18.)  Two  stakes,  yi  and  /?,  are  on  opposite  sides  of  a  stream;  a 
third  stake,  C,  is  set  62  ft.  from  A ;  the  angles  ACB  and  CAB  are 
found  to  be  50°  3'  5"  and  61^'  18'  20"  respectively.  How  long  is  a 
rope  connecting  A  arid  B} 

(19.)  To  find  the  distance  between  two  inaccessible  mountain-tops, 
A  and  B,  of  practically  the  same  height,  two  points,  C  and  D,  are 
taken  one  mile  apart.  The  anj^le  CD  A  is  found  to  be  88°  34',  the 
angle  DC  A  is  63°  8',  the  angle  CDB  is  64°  27',  the  angle  DCB  is  87°  9'. 
What  is  the  distance? 

(20.)  Two  islands,  jff  and  C  are  distant  5  and  3  miles  respectively 
from  a  light-house,  .<4,  and  the  angle  BAC  is  33°  11';  find  the  dis- 
tance between  the  islands. 

(21.)  Two  points,  A  and  B,  are  visible  from  a  third  point  C,  but 
not  from  each  other;  the  distances  AC,  BC,  and  the  angle  ACB  were 
measured,  and  found  to  be  1321  ft.,  1287  ft.,  and  61°  22'  respectively. 
Find  the  distance  AB. 

(22.)  Of  three  mountains.  A,  B,  and  C,  B  is  directly  north  of  C  5 
miles,  ^  is  8  miles  from  Cand  11  from  B.     How  far  is  ^  south  of  B} 

(23.)  From  a  position  215.75  ft.  from  one  end  of  a  building  and 
198.25  ft.  from  the  other  end,  the  building  subtends  an  angle  of 
53°  37'  28'';  find  its  length. 

(24.)  If  the  sides  of  a  triangle  are  372.15,  427.82,  and  404.17  ;  find 
the  cosine  of  the  smallest  angle. 

(25.)  From  a  point  3  miles  from  one  end  of  an  island  and  7  miles 
from  the  other  end,  the  island  subtends  an  angle  of  33°  55'  15";  find 
the  length  of  the  island. 

(26.)  A  point  is  13581  in.  from  one  end  of  a  wall  12342  in.  long,  and 
10025  i"-  from  the  other  end.  What  angle  does  the  wall  subtend  at 
this  point.'' 

(27.)  A  straight  road  ascends  a  hill  a  distance  of  213.2  ft.,  and  is  in- 


THE  OBLIQUE   TRIANGLE  53 

clined  12°  2'  to  the  horizontal ;  a  tree  at  the  bottom  of  the  hill 
subtends  at  the  top  an  angle  of  10^  5'  16".  Find  the  height  of  the 
tree. 

(28.)  Two  straight  roads  cross  at  an  angle  of  37°  50'  at  the  point  A  ; 
3  miles  distant  on  one  road  is  the  town  B,  and  5  miles  distant  on  the 
other  is  the  town  C.     How  far  are  B  and  C apart?  \ 

(29.)  Two  stations,  A  and  B,  on  opposite  sides  of  a  mountain,  are 
both  visible  from  a  third  station,  C\  ^C  1=11.5  niiles,  i)'C  =  9.4  miles, 
and  the  angle  ACB  =z  59°  31'.     Find  the  distance  from  A  to  B, 

(30.)  To  obtain  the  distance  of  a  battery.  A,  from  a  point,  B,  of  the 
enemy's  lines,  a  point,  C,  2,7 ~-7  yards  distant  from  A  is  taken  ;  the  an- 
gles ACB  and  CAB  are  measured  and  found  to  be  79°  53'  and  74°  35' 
respectively.     What  is  the  distance  AB} 

(31.)  A  town,  ^,  is  14  miles  due  west  of  another  town,  A.  A  third 
town,  C,  is  19  miles  from  A  and  17  miles  from  B.  How  far  is  C  west 
oiA} 

(32.)  Two  towns,  A  and  B,  are  on  opposite  sides  of  a  lake.  A  is 
18  miles  from  a  third  town,  C,  and  B  is  13  miles  from  C\  the  angle 
ACB  is  13°  17'.     Find  the  distance  between  the  towns  A  and  B. 

(33.)  At  a  point  in  a  level  plane  the  angle  of  elevation  of  the  top 
of  a  hill  is  39°  51',  and  at  a  point  in  tlie  same  direct  line  from  the  hill, 
but  217.2  feet  farther  away,  the  angle  of  elevation  is  2&^  53'.  Find 
the  height  of  the  hill  above  the  plane. 

(34.)  It  is  required  to  find  the  distance  between  two  inaccessi- 
ble points,  A  and  B.  Two  stations,  C  and  D,  2547  ft.  apart,  are 
chosen  and  the  angles  are  measured  ;  they  are  ACB^27°  21',  BCD 
=33°  14.  BDA=iS°  17',  and  ADC=si°  23'.  Find  the  distance  from 
AtoB. 

(35.)  Two  trains  leave  the  same  station  at  the  same  time  on  straight 
tracks  inclined  to  each  other  21°  12'.  If  their  average  speeds  are  40 
and  50  miles  an  hour,  how  far  apart  will  they  be  at  the  end  of  the  first 
fifteen  minutes  ? 

(36.)  A  ship,  A,  is  seen  from  a  light-house,  B;  to  determine  its  dis- 
tance a  point,  C,  300  ft.  from  the  light-house  is  taken  and  the  angles 
BCA  and  CBA  measured.  If  BCA  =  ioS°  34'  and  CBA  =65°  27',  what 
is  the  distance  of  the  ship  from  the  light-house.' 


54 


PLANE    TRIGONOMETRY 


(37,)  Prove  that  the  radius  of  the  inscribed  circle  of  a  triangle  is 
equal  to  a  %\x\\B  sin^Csec^^. 


Hint. — Draw  OB,  OC,  and  the  perpendicular  OD. 
OB  and  C>C  bisect  the  angles  B  and  C  respectively,  and  OD=r. 
a=BD  +  DC=r{cotiB  +  cotiC). 

sin  I C  cos  ^^  + cos  |Csin  ^B 


coti^  +  cotiC=- 


siniB  sin  ^C 
sin  HB  +  C)  cos  i  A 


Hence 


sin  ^  A' sin  ^C      sin^//sin|C 
sin  1^  sin^C 


cosi^ 


-  =  a  sin  ^  ^  sin  ^  C  sec  ^A, 


CHAPTER  V 

CIRCULAR  MEASURE— GRAPHICAL  REPRESENTATION 

CIRCULAR    MEASURE 

S3,  The  length  of  the  semicircumference  of  a  circle  is 
wR  (7r  =  3.i4i59  +  ) ;  the  angle  the  semicircumference  sub- 
tends at  the  centre  of  the  circle  is  i8o°.  Hence  an  arc 
whose  length  is  equal  to  the  radius  will  subtend  the  angle 

i8o°       ,  .  ,      .      ,  .  ,        r      .       , 

;   this  angle  is  the  unit  angle   of  circular   measure, 


and  is  called  a  radian. 


TV  R 


If  the  radius  of  the  circle  is  unity,  an  arc  of  unit  length 
subtends  a  radian  ;  hence  in  the  unit  circle  the  length  of  an 
arc  represents  the  circular  measure  of  the  angle  it  subtends. 

Thus,  if  the  length  of  an  arc  is      ,  it  subtends  the  angle  -  radians. 


Since  one  radian 


1 80° 


,  we  have 


90  =      radians, 
i8o°  =  7r  radians, 


56  PLANE    TRIGONOMETRY 

270  =-  radians, 

'  2 

360°  — 27r  radians,  etc. 
The  value  of  a  radian  in  degrees  and  of  a  degree  in  radians  are  ; 
I  radian  =  57.29578°, 

=  57°  1 7' 45". 
i°=. 0174533  radian. 
In  the  use  of  the  circular  measure  it  is  customary  to  omit  the  word  radian ; 

TT  TV 

thus  we  write  -  ,  7r,  etc.,  denoting  -  radians,  re  radians,  etc.      On  the  other 
2  "2 

hand,  the  symbols  °  '  "  are  always  printed  if  an  angle  is  measured  in  degrees, 

minutes,  and  seconds ;  hence  there  is  no  confusion  between  the  systems. 

EXERCISES 

(I.)  Express  in  circular  measure  30°,  45°,  60°,  120°,  135°,  720°,  990°. 
(Take  7r  =  3.i4i6.) 

(2.)  Express  in  degrees,  minutes,  and  seconds  the  angles  o'  —  >  -  •    • 

(3.)  What  is  the  circular  measure  of  the  angle  subtended  by  an  arc 
of  length  2.7  in.,  if  the  radius  of  the  circle  is  2  in..''  if  the  radius  is 
5  in.  ?       . 

54:,  The  following  important  relations  exist  between  the 
circular  measure  x  of  an  angle  and  the  sine  and  tangent  of 
the  angle. 

TT 

(i .)  If  X  is  less  than  -,  sin x<x < tan x. 


O  S 

Draw  a  circle  of  unit  radius. 

By  Geometry,        SP Kd^xc  AP <AT. 
Hence  sin  .«•  <  ;r  <  tan^. 


CIRCULAR  MEASURE  57 

stn  X          tcin  X 
(2.)  As  X  approaches  the  limit  o, and approach 

the  limit  i.  ■ 

Dividing  sin  x  <x  <  tan  x  by  sin x,  we  obtain 

X  I 

i<-^ < 


sm;tr      cos;tr 

_           .                                    sin  ;t'      cos;tr 
Inverting,  i> > -^ . 

As  X  approaches  the  hmit  o,  cos;ir  approaches  the  length 

of  the  radius,  that  is,  i,  as  a  Hmit. 

sin  X 
Therefore, approaches  the  hmit  i. 

T-w-  -J-              sin;tr  ,  ,      . 

Uividmg  I  > >cos;tr  by  cos;r,  we  obtam 

I         tan  X 

> ^>i. 

cos  X  X 

As  X  approaches  the  Hmit  o,  cos;ir  approaches  the  Hmit  i ; 

hence  approaches  the  Hmit  i. 

cos  X 

tcin  jir 
Therefore,  approaches  the  H'mit  i. 


PERIODICITY   OF  THE   TRIGONOMETRIC   FUNCTIONS 

55*  The  sine  of  an  angle  x  is  the  same  as  the  sine  of 
(^+360°),  (;ir  +  720°),  etc.— that  is,  of  (;ir  +  2«7r),  where  n  is 
any  integer. 

The  sine  is  therefore  said  to  be  a  periodic*  function,  hav- 
ing the  period  360°,  or  27r. 

The  same  is  true  of  the  cosine,  secant,  and  cosecant. 

*  If  a  function,  denoted  by/(;c),  of  a  variable  x,  is  such  XhiX.  f{x-\-k^=.f{x) 
for  every  value  of  x,  k  being  a  constant,  the  function  f{x)  is  periodic ;  if  /i  is 
the  least  constant  which  possesses  this  property,  k  is  the  period  oi  f(x). 


58 


PLANE    TRIGONOMETRY 


The  tangent  of  an  angle  x  is  the  same  as  the  tangent  of 
{x-\- 1 80°),  (;ir+36o°),  etc. — that  is,  of  (;ir+«7r),  where  71  is  any 
integer. 

The  tangent  is  therefore  a  periodic  function,  having  the 
period  180°,  or  tt. 

The  same  is  true  of  the  cotangent. 


GRAPHICAL    REPRESENTATION 

50,  On  the  line  OX  lay  off  the  distance  OA{=x)  to  rep- 
resent the  circular  measure  of  the  angle  x.  At  the  point  A 
erect  a  perpendicular  equal  to  sin  ;r.  If  perpendiculars  are 
thus  erected  for  each  value  of  x,  the  curve  passing  through 
their  extremities  is  called  the  sine  curve. 

If  sinjT  is  negative,  the  perpendicular  is  drawn  downward. 


In  a  similar  manner  the  cosine,  tangent,  cotangent,  secant, 
and  cosecant  curves  can  be  constructed. 

+1 


-1 


/ 

1 
1 

1 

\ 

* 

/ 

1     ^S^ 

1 

1 

\ 

0 

8 

A, 

V 

1   ^/^ 

X2  7r 

fs^^ 

} 

Sine  Curve 


Cosine  Curve 


GRAPHICAL   REPRESENTATION  59 

I 


Cotangent  Curve 


6o 


PLANE   TRTGONOAfETRY 


O 


O 


%T- 


2k 


2J^i- 


SECANT   CURVE 


If  the  distances  on  OX  axe  measured  from  O'  instead  of 
O,  we  obtain  from  the  secant  curve  the  cosecant  curve. 

In  the  construction  of  the  inverse  curves  the  number  is 
represented  by  the  distance  to  the  right  or  left  from  O; 
the  circular  measure  of  the  angle  by  the  length  of  the  per- 
pendicular erected. 

All  of  the  preceding  curves,  except  the  tangent  and  co- 
tangent curves,  have  a  period  of  27r  along  the  line  0X\  that 
is,  the  curve  extended  in  either  direction  is  of  the  same 
form  in  each  case  between  27r  and  47r,  47r  and  67r, — 27r  and 
o,  etc.,  as  between  o  and  27r,  while  the  corresponding  inverse 
curves  repeat  along  the  vertical  line  in  the  same  period. 
The  period  of  the  tangent  and  cotangent  curves  is  tt. 


GRAPHICAL   REPRESENTATION  61 


-1  0         +1 


-1  0         + 


INVERSE    SINE   CUKVK 


INVERSE    COSINE   CUKVK 


■2  -1  0  +1 


INVERSE   TANGENT   CURVE 


62 


PLANE    TRIGONOMETRY 


2^'r 


8T 


%t: 


-3         -2  -I  0         +^         +2         +3 

INVERSE   SECANT 


CHAPTER   VI 

COMPUTATION    OF    LOGARITHMS   AND   OF  THE   TRIG- 
ONOMETRIC FUNCTIONS -DE   MOIVRE'S  THEOREM 
—HYPERBOLIC   FUNCTIONS 

57»  A  convenient  method  of  calculating  logarithms  and 
the  trigonometric  functions  is  to  use  infinite  series.  In 
works  on  the  Differential  Calculus  it  is  shown  that 

Or»2  ^v*3  rn^A 

Iog,(l  +  a7)  =  x- 2-+-3 -^+ .  .  .  (i) 

^y»3  /Trt5  />»^ 

.  uM.^  ^Mj  *My  . 

glna5  =  a;-^+-^,-yy+  ...  *  (2) 

rg*^  nt*^  nf*6 

C08X=l-^,+^-^,  +  ...  (3) 

Another  development  which  we  shall  use  later  is 

/nf  rtf*^  /Tr*3  /v»4 

.^  •*/  tA/  t/U  mfU 

C-  =  l  +  j^  +  ^  +  3-,  +  j-,+  ...  (4) 

where  ^=2.7182818  ...  is  the  base  of  the  Naperian  system 
of  logarithms. 

58.  The  series  (i)  converges  only  for  values  of  x  which  satisfy  the 
inequality  —  i<xSi.  The  series  (2),  (3),  and  (4)  converge  for  all 
finite  values  of  x. 

It  is  to  be  noted  that  the  logarithm  in  (i)  is  the  Naperian,  and  the 
angle  x  in  (2)  and  (3)  is  expressed  in  circular  measure. 

*  3!  denotes  1x2x3;  41  denotes  1x2x3x4,  etc. 


64  PLANE   TRIGONOMETRY 

COMPUTATION    OF   LOGARITHMS 

60*  We  first  recall  from  Algebra  the  definition  and  some 
of  the  principal  theorems  of  logarithms. 

The  logarithm  to  the  base  a  of  the  number  m  is  the  number  x 
which  satisfies  the  equation, 

^^  =  m. 

This  is  written  x  =  log^  m. 

The  logarithm  "of  the  product  of  two  numbers  is  equal  to  the  sum 
of  the  logarithms  of  the  numbers. 

Thus  1  oga  mn  —  log^  m  + 1  og^  «. 

The  logarithm  of  the  quotient  of  two  numbers  is  equal  to  the  log- 
arithm of  the  dividend  minus  the  logarithm  of  the  divisor. 

Thus  log^-  =  log„w  — log^«. 

n 

The  logarithm  of  the  power  of  a  number  is  equal  to  the  logarithm 
of  the  number  multiplied  by  the  exponent. 

Thus  \oga  ffi^=p  log„  m. 

To  obtain  the  logarithm  of  a  number  to  any  base  a  from  its  Na- 
perian  logarithm,  we  have 

log« '«  =  -; =  M„  log,  m, 

log^a 

where  M^  = . ;  M-  is  called  the  modulus  of  the  system. 

log^a 

60*  We  proceed  now  to  the  computation  of  logarithms. 

The  series  (i)  enables  us  to  compute  directly  the  Naperian 

logarithms  of  positive  numbers  not  greater  than  2. 

■I 
Example. — To  compute  log*-  to  five  places  of  decimals. 

Substitute  -  for  x  in  (i): 
2 

log---=log,  {l  +  -)= T,^---\ •  -.+    ... 

*"?  2  '''  \  2/       2      2     2''       3     2-*       4     2"' 

If  the  result  is  to  be  correct  to  five  places  of  decimals,  we  must  take  enougli 
terms  so  that  the  remainder  shall  not  affect  the  fifth  decimal  place.     Now  we 


COMPUTATION  OF  LOGARITHMS 


65 


know  by  Algebra  that  in  a  series  of  which  the  terms  are  each  less  in  numerical 
value  than  the  preceding,  and  are  also  alternately  positive  and  negative,  the  re- 
mainder is  less  in  numerical  value  than  its  first  term.  Hence  we  need  to  take 
enough  terms  to  know  that  the  first  term  neglected  would  not  affect  the  fifth 
place. 

Positive  terms 


-   =0.5000000 


—.  =   .0416667 

I 


=  .0062500 


—  =  .OOIII61 

2' 


—  =  .0002170 

2^ 


-=  .0000444 

II   2" 


—  •  -T-„=   .0000094 
13    2'^       

.5493036 


Negative  terms 

I 

2 

I 

—  =0.1250000 

I 

4 

I 
2*^ 

.0156250 

I 
6 

I 

26~ 

.0026042 

I 
8 

I 

78  ~ 

.0004883 

I 
10 

I 

.0000977 

I 
12 

I 

^2- 

.0000203 

I 
14 

I 

.0000044 

■1438399 


Subtracting  the  sum  of  the  negative  from  the  sum  of  the  positive  terms,  v 
obtain 


log^-=  •4054637- 


bra, 


Denote  the  sum  of  the  remaining  terms  of  the  series  by  R. 


15   2' 


< 


The  error  caused  by  retainir- 
less  than  .0000006.     Hence 
the  result  is  correct  to  five  d 

Gl»  As  remarke 
calculate  directly  • 
but  it  can  be  read  . 
us  the  logarithm  o 

Replacing  :v  by 
5 


66  PLANE   TRIGONOMETRY 


\0ge  {\  —  x)=—X 

234 


This  series  converges  for  —x'^x-Cx, 
Subtracting  this  from  (i),  we  obtain 

log,  (i+^)-log,  (i_^)  =  log,  f^^j 

I       x^    x"    x'  \ 

which  converges  for  —  i  <  ^  <  i . 

——1,  we  see  that  y  passes  from  o  to  00  as  ;c 

passes  from  —  i  to  +1  ;  hence,  if  we  make  this  substitution  in 
(5),  we  get  a  series 

which  converges  for  all  positive  values  oi  y,  and  therefore  enables 
us  to  compute  the  Naperian  logarithm  of  any  number. 

From   (5)  we   can   get   another  series  which  is   useful :.  put 

•  then,  as      — =c -,  equation  (5)  gives  us 

K2y^W      7*^^^ 
!«  knowa.     It  con- 
an  2,  and  hence 


w'lerc  >'  values  of  >'.     Hence, 


'e  directly 
■thers  can 


eriTis  ao  tiMlt 


COMPUTATION  OF  LOGARITHMS 


67 


Thus,  to  obtain  the  logarithms  of  the  integers  up  to  10, 
we  need  to  compute  by  series  only  the  logarithms  of  the 
numbers  2,  3,  5,  and  7. 

(For  4=2'-,  b—2 .  3,  8  =  2',  9=3',  10=2  .  5,  and  log  i=o.) 
In  this  case  we  are  computing  the  logarithms  of  successive  integers,  and 
should  therefore  use  (7). 

63.  Example. — Compute  the  Naperian  logarithms  of  2,  3,  4,  and  5. 


loG,2  =  2(  -  +  -  .  — + 

\3     3    3'     5 


3     3    3'     5    3' 


'a'- 


III 

3'     9     3 


)■ 


-=•3333333 


:=. 0123457 


—=.0008230 
-,  =  .0000653 


9    3 


^=.0000056 
.3465729 


Denote  the  sum  of  the  remaining 
terms  of  this  series  by  R. 
Then,  by  Algebra, 

II  3"  i-i 
or   /('<  .000000573. 

The  error  caused  by  not  retaining 
more  places  of  decimals  in  the  pre- 
ceding column  is  less  than  .0000005. 

Hence,  the  total  error  is  less  than 
.00000165. 


log^2  =  . 6931458 
Remark. — We  should  get  the  same  series  if  we  were  to  use  (6). 


log,  3  =  log,2  +  2Q  +  ^ 


-  +  - 
3    5'     5 


5 

.2000000 

I 

3' 

1 
5'~ 

.0026667 

I 
5  " 

I 

.0000640 

I 
7  ' 

I 

.0000018 

.2027325 

2 

.4054650 

Add  log 

.2  = 

.6931458 

log*  3= 1. 0986108 


5'     7    5\ 


A  <  -  •  —z 

9    5' 


or  R<.  .00000006. 

Noting  the  errors  in  the  pre- 
ceding column  and  in  log,  2,  we 
see  that  the  total  error  is  less  than 
.00000217. 


68  PLANE    TRIGONOMETRY 

Remark. — If  we  were  to  use  (6)  to  compute  log,.  3,  we  should  have 

-— [j+-;(0'+;G)'+KO'-]- 

This  series  converges  much  more  slowly  than  the  above,  since  its 
terms  are  multiples  of  powers  of  \,  while  the  terms  of  the  above  are 
the  same  multiples  of  powers  of  \.  Thus,  we  should  be  obliged  to 
use  eight  instead  of  four  terms  to  have  the  result  correct  to  five 
places. 

log,  4  =  2  log,  2  =  1. 3862916. 

log.5  =  log,4  +  2Q  +  i.^+i.^,+  ...). 
or  log*  5  =  1-60944. 

G4:,  Proceeding  in  like  manner,  we  may  calculate  any  number  of 
logarithms. 

The  following  table  gives  the  Naperian  logarithms  of  the  first  ten 
integers: 


log* I =     -00000 

log,  2=  .69315 
If'g,  3  =  1-09861 
I'^g,  4  =1-38629 
log,  5  =  1-60944 


log,  6  =1.79176 

log,  7  =  I-9459I 

log,  8  =  2.07944 

log, 9  =  2. 19722 

log,  10  =  2.30259 


The  common  logarithm  of  any  number  may  be  found  by  multiply- 
ing its  Naperian  logarithm  by  Mio=. 43429448.  §59 

Thus  log,„  5  =  log,  5  X  .43429448  =  .69897. 

60.  Remark. —  If  a  table  of  logarithms  were  to  be  computed,  the 
theory  of  interpolation  and  other  special  devices  would  be  employed. 

COMPUTATION  OF  TRIGONOMETRIC   FUNCTIONS 

Sin  X  cos  X 

06*  Since  tan;r=-^^ ,cot.ar=-^ — '-,  etc.,  the  computa- 

cos.r  sin;»r 

tion  of  all  the  trigonometric  functions  depends  upon  that  of 
the  sine  and  cosine ;  thus  the  developments  (2)  and  (3)  suf- 
fice for  all  the  trigonometric  functions.     Further,  since  the 


COMPUTATION  OF   SINES  AND   COSINES  69 

sine  or  cosine  of  any  angle  is  a  sine  or  cosine  of  an  angle 

"::— ,  it  is  never  necessary  to  take  x  greater  than  -  in  the 
<-4  4 

series  (2)  and  (3).  §  16 

Since  -  =0.785398  ...-<—,  these  series  converge  rapidly  ;  in  fact, 
4  10 

—  =  .000003   does   not  affect  the  fifth  decimal   place,  and   —  the 
9!  11! 

seventh. 

07'  Remark. — In  the  systematic  computation  of  tables  we  should 
not  calculate  the  functions  of  each  angle  from  the  series  independent- 
ly. We  should  rather  make  use  of  the  formulas  (25)  and  (27)  of  §  38, 
thus  obtaining 

sin nx :=2  cos x  sin  {n  —  \)x  —  sin  {n  —  2) x, 
cos nxz=.2  cos X  cos  («  —  \)x  —  cos {n  —  2) x. 

If  our  tables  are  to  be  at  intervals  of  i',  we  should  calculate  the 
sine  and  cosine  of  i'  by  the  series.  The  above  expressions  then  en- 
able us  to  find  successively  the  sine  and  cosine  of  2',  3',  4',  etc.,  till  we 
have  the  sine  and  cosine  of  all  angles  up  to  30°  at  intervals  of  i'. 

To  obtain  the  sine  and  cosine  of  angles  from  30°  to  45°  we  should 
make  use  of  these  results  by  means  of  the  formulas 
sin  (30° +j)  =:cos^  —  sin  (30° — /), 
cos  (30°+_y)  =  cos  (30^ — y)  — sin_y. 

08*  To  employ  series  (2)  and  (3)  in  computing  the  sine 
and  cosine  we  must  first  convert  the  angle  into  circular 
measure. 

To  do  this  we  recall  that 

1°  =  .017453293,     I '  =  .0002908882,     I  "  =  .000004848 1 37. 

Example. — To  compute  the  sine  and  cosine  of  12°  15'  39". 

12°=  .209439516 
15'  =.004363323 
39"  =  .000189076 
12°  15'  39"  =  .213991915  in  circular  measure. 


70 


PLANE   TRIGONOMETRY 


%\vix-=.x 1 -  — 

3!      5! 

jr=.2i399i9 


— =.0000037 

.2139956 
x" 
subtract  — =.0016332 
3! 

sin  jr=. 2123624 

Correct  to  five  decimal  places. 


cosjr=i -H 

2  !      4  ! 

1  =  1.0000000 
— =  .0000874 

1.0000874 

x^ 
subtract  — f=  .0228963 


cosjr=  .9771911 
Correct  to  five  decimal  places. 


DE   MOIVRE'S  theorem 
09.  In  Algebra  we  learn  that  the  complex  number 

a=a4-/3\/^  =  a+i3/  (8) 

'may  be  represented  graphically  thus : 


Take  two  lines,  OX  and  OY,  at  right  angles  to  each  other. 
To  the  number  a  will  correspond  the  point  A,  whose  dis- 
tances from  the  two  lines  of  reference  are  /9  and  a  re- 
spectively. 

This  geometrical  representation  shows  at  once  that  we 
can  also  write  a  in  the  form 

a  —  r  (cos  5  -|- 1  sill  5).  (9) 

70.  From  Algebra  we  recall  the  definition  of  the  sum  of  the 
complex  numbers  a  —  a-\-i\^  and  b  =  y-\-il\  namely 
a  +  /?=:a  +  y-|-/(/3-ra). 
Subtraction  is  defined  as  the  inverse  of  addition,  so  that 
a-b-a  —  y-^i{l^-l). 


DE  MOIVRE'S   THEOREM  71 

Multiplication  is  most  conveniently  defined  when  a  and  b  are 
written  in  form  (9).     If 

a  —  r  (cos  ^4-/  sin  ■?)  and  b—s  (cosf+i  sin^), 
their  product  is  defined  by  the  equation 

ab  —  rs  [cos(5  +  ^)4-/ sin(^-}-^)].  (10) 

Division  is  defined  as  the  inverse  of  multiplication,  so  that 

-  =  -  [cos  (-^  —  0)  -f-  /  sin  (-i? — ^)]. 

Finally,  we  recall  that  in  an  equation  between  complex  numbers, 

we  have  a  =  y,    fi  —  Z.  (11) 

7 1»  Consider  the  different  powers  of  the  complex  number 

a:=cos  ^-\-i  sin^. 
By  (10)  we  have 

;(;'=:  (cos  54-  /  sin  ■?)  (cos  ^-\-i  sin  -&), 
—  cos2^-\-i  sin  2^. 
x'  —  x^ .  x  =  (cos2S4-^  sin  2B)  (cos5-|-/  sin  ^), 
=  cos  3^  -f-  /'  sin  3-^. 
And,  in  general,  for  any  integer  n, 

;«:''  =  (cos  B-{-i  sin  ^)«=cos  n^-\-z  sin  «•&. 
From  this  equation  we  have  De  Moivre's  Theorem,  which 
is  expressed  by  the  formula 

(co9-&+tsin5)"=(co8n^+islnn-&).  (12) 

72.  An  interesting  application  of  De  Moivre's  Theorem 
is  the  expansion  of  sin  fix  and  cos  nx  in  terms  of  sin  x  and 
cos;r.  Expanding  the  left-hand  side  of  (12)  by  the  bino- 
mial theorem,  and  substituting  x  for  S',  we  have 

fi(fi I  ^ 

cos«j:+/ sin«;c  =  cos''A:-|-«  cos"~'j:  (/ sin  ^)  -{-— — j — cos^^^j; 

/.    .      N«     «.(«  — 1)(«  — 2)         ,    ,      ,.  .      ,, 

(»  sin xy -\ 5^ ^^ ^  cos" -^x(i sin xY  -\-  .  .  . 

3  ! 


72  PLANE    TRIGONOMETRY 

or 

.    .             /              nin-i)  .  ,  \ 

co%nx-\-t  %\x\nx—\zo%*^ X ^^ — j —  cos"""* a:  sln^3c-^-  .  .  .] 

[«-.        •          n  (n—i)  (n  —  2)        «    ,       .   ,  1 

;/  cos"    '  X  sin^ ^^ '— — — ^  cos"~3;c  sin  a: 4-  .... 
3  !  J 

Equating  real  and  imaginary  parts,  as  in  (ii),  we  have 

cos;/.x=cos".:v ^ — j — ■'^  cos"~*jt:  sin\x:-|-  .  .  .  (13) 

«  — ,       •  «(«  — 1)(«  — 2)  ,   ,  ,     , 

sm;/x  =  «cos"     'A:sni;c ^= — cos"'"J.xsm  a:+.  . ,  (\/CS 

3  !  I  \  -r/ 

Example. — «  =  5. 

cos  5-r  =z cos*  X  —  10  cos'-r  sin'jr+s  cos .r  sin*  j:. 

sin  5-r  =  5  cos* x  %\nx—\o  cos"  x  sin' .r  +  sin* x. 

THE   ROOTS   OF   UNITY 

75.  We  find  another  application  of  De  Moivre's  Theorem 

in  obtaining  the  roots  of  unity.     The  «*•>  roots  of  unity  are 

by  definition  the  roots  of  the  equation 

;r«=i. 

Every  equation  has  n  roots  and  no  more ;  hence,  if  we 

can  find  n  distinct  numbers  which  satisfy  this  equation  we 

shall  have  all  the  «'*>  roots  of  unity. 

Consider  the  h  numbers 

27rr      .    .     2irr 

Xr  —  zo% \-t  sin , 

n  n 

r=o,  1,  2,  .  .  .  «— I. 

Geometrically  these  numbers  are  represented  by  the  n 
vertices  of  a  regular  polygon.  They  are,  therefore,  all  dif- 
ferent. We  shall  see  now  that  they  are  precisely  the  «'*> 
roots  of  unity. 

In  fact,  we  have  by  (12), 


_        /  2-irr        .     .       27rr\'» 

.r'l^lcos \-i  sin j  , 


THE   ROOTS   OF  UNITY  73 

(27rr\       .    .     /       2xr\ 

=  cos  2Tr-f/sit)  27r;-, 

=  14-/.  0  =  1. 
Therefore  x^  is  one  of  the  roots  of  unity. 

Thus  the  cube  roots  of  unity  are  represented  by  the  points  A,  P, 
and  <2  of  the  following  figure.     In  the  figure  0A=.  i,  angle  AOP^ 

—  =  120°,  angle  AOQ  =  — =  2^0°;   that  is,  the  circumference  is  di- 
3  ''  "^       3 

vided  into  three  equal  parts  by  the  points  A,  P,  and  Q.    Then  OD  =  ^, 

and  DP=:DQ  =  ^y/2-     Hence  we  see  from  the  method  of  represent- 
ing a  complex  number  given  above  that  ^  represents -f-i.-^  represents 

—  k  +  ^^Vi'  Q  represents  —^  —  i^i- 


EXERCISES 

74.  (I.)  Express  sin  4jc  and  cos4;r  in  terms  of  sin;r  and  cos;r. 

(2.)  Express  sin  6^  and  cos6x  in  terms  of  sin  j:  and  cos  jr. 

(3.)  Find  the  six  6'*'  roots  of  unity. 

(4.)  Find  the  five  5*  roots  of  unity. 


THE    HYPERBOLIC   FUNCTIONS 
7S.  The  hyperbolic  functions  are  defined  by  the  equations 


sinh  X  = 


(15) 


cosli  oc  = , 


(16) 


in  which  sinh  x  and  cosh  x  denote  the  hyperboh'c  sine  and 


74  PLANE   TRIGONOMETRY 

hyperbolic  cosine  of  x  respectively.  These  functions  are 
called  the  hyperbolic  sine  and  cosine  on  account  of  their 
relation  to  the  hyperbola  analogous  to  the  relation  of  the 
sine  and  cosiile  to  the  circle.  A  natural  and  convenient 
way  to  arrive  at  the  hyperbolic  functions  and  to  study  their 
properties  is  by  using  complex  numbers  in  the  following 
manner.  The  series  (2),  (3),  and  (4)  give  the  value  of  sin  x^ 
cos-f,  and  e''  for  every  real  value  of  x.  These  series  also 
serve  to  define  sin;tr,  cos;ir,  and  ^^for  complex  values  of  x. 
In  the  more  advanced  parts  of  Algebra  it  is  shown  that 
the  following  fundamental  formulas  which  we  have  proved 
only  for  a  real  variable, 

sin  {x-\-y)z=s\xix  cos^y-l-cosx  sin 7,  (17) 

cos  (^4-_>')=:Cos^  cosj'  — siuj;  sin_y,  (18) 

e'^ry^gx^y^  (19) 

hold  unchanged  when  the  variable  is  complex. 

This  fact  enables  us  to  calculate  with  ease  sinx-,  cosa',  and 
r^  for  any  complex  value  of  the  variable. 

In  so  doing  we  are  led  directly  to  the  hyperbolic  func- 
tions. At  the  same  time  a  relation  between  the  trigono- 
metric and  hyperbolic  functions  is  established  by  means  of 
which  the  formulas  of  Chapter  III.  can  be  converted  into 
corresponding  formulas  for  the  hyperbolic  functions. 

Taking  x  and  y  real  and  replacing  y  in  (17),  (18),  and  (19)  by 

iy,  we  get 

sin  (:c +  />')=: sin  jc  cos//4-cos:!C  sin  iy, 

cos  (.T+Zy)  =  cos  X  cos  (y— sin  x.  sin  iy, 

e'^'y=e^/y. 

Thus  the  calculation  of  these  functions  when  the  variable 
is  complex  is  made  to  depend  upon  the  case  where  the  vari- 
able is  a  pure  imaginary. 


HYPERBOLIC  FUNCTIONS  75 

If  we  replace  x  by  ix  in  series  (4)  we  obtain 

.      iixf     {ixf     iixy 


/       ,r'     x'     x'  \ 

.{        x'      x"     x'  \ 

+Ax -I--  — +  . . .  . 

V       3 !     5 !     7  / 


A  comparison  with  series  (2)  and  (3)  shows  that  these  two 
series  are  cos;i-  and  sin -r  respectively;  hence  the  important 
formula  due  to  Euler — 

e'^=  cos  ic-|-*  sin  .x.  (20) 

This  enables  us  to  calculate  ^'-^  from  sin;r  and  cos;ir  when 
ix  is  a  pure  imaginary ;  that  is,  when  x  is  real. 

To  find  sin  ix  and  costx  replace  x  in  (20)  by  ix\  we  obtain 

<f~-^=cos /;c  +  / sin /x,  (21) 

Again  replacing  x  by  —ix  in  (20),  we  obtain 

e^ = cos  ix  —  /  sin  ix.  (2  2) 

The  sum  and  difference  of  (21)  and  (22)  give 

COS«iC=     —       =:COShiC,  (23) 

sin*J7  = =  i8inli^.  (24) 

If  we  compute  the  value  of  e^  by  the  aid  of  series  (4)  for 
a  succession  of  values  of  x,  we  find  that  sinh;r  and  cosh,r 
are  represented  by  the  curves  on  page  'j6. 

The  system  of  formulas  belonging  to  the  hyperbolic  func- 
tions is  obtained  from  those  of  the  trigonometric  functions 
by  using  (23)  and  (24).  This  shows  that  for  every  formula 
in  analytic  trigonometry  there  exists  a  corresponding  for- 
mula in  hyperbolic  trigonometry  which  we  get  by  this  sub- 


76 


PLANE   TRIGONOMETRY 


stitution.  In  the  examples  which  follow,  this  method  is 
used  to  obtain  important  formulas  in  hyperbolic  trigonome- 
try. 

Replacing  x  by  —ix  in  (23)  and  (24),  we  get 


COSO?: 


sin  X  = 


alX^t>  —  tX 


(25) 
(26) 


which  are  formulas  frequently  used. 


Example. — sinh  (.r  -{-y)  ■=  —  /  sin  z{x -\-y), 

=  —  /  [sin  ix  cos  iy  -\-  cos  ix  sin  iy"], 

=  —i  [/sinh  X  cosh_y-f /cosh  x  sinhj], 

=  sinh  X  cosh/  -j-  cosh  x  sinh/. 

Example. — sinh  x-{-%\n\\y  =  —  /(sin  ix  -\-  sin  iy), 

=.  —  /  2  sin  J  i(.r-\-y)  cos  ^  i{x — y), 
=  2  sinh  i  {x  -\-y)  cosh  ^  (-r  — /). 


sinTi  cc 


cosh  X 


HYPERBOLIC  FUNCTIONS  77 

EXERCISES 
76.  (I.)  Prove  sinho=o,     cosh 0=1. 
(2.)  Prove  sin h^7r/  =  z,     cosh  ^7r/=o. 
(3.)  Prove  sinh7rz'  =  o,     cosh7r/=  — i. 

Prove  that 

(4.)  sin( — z".r)  =  — sin/^.  :  .!  '>  -:  ,  ;..; 

(5.)  cos  {—ix)  =  cos  2X. 

(6.)  sinh(— jr)  =  —  sinh  jr. 

(7.)  cosh( — .r)  =  cosh;ir. 

Remark. — The  hyperbolic  tangent,  cotangent,  secant,  and  cosecant 

are  defined  by 

^     ,  sinhjf  ,  cosh^ 

tanh;r=: — -,  cothjir  =  -7— ; — . 

cosh;r  sinhjr 

sech  .r  = ; — .  csch  x  =  ■ 


coshjf  sinh-r 

Prove  that 

(8.)  tan  (z'x)  —  i  tanh  x. 

(9.)  coth  {—x)  =  —  coth  X. 

(10.)  sech  {—x)  —  sech  x. 

(II.)  cosh'x— sinh'.r^j. 

(12.)  sech^r-|-tanh*.r=  I. 

(13.)  coth^r  —  csch*.r=i. 

(14.)  sinh(x — J/)  =sinh;ir  cosh^  — cosh.r  sinhj/. 

(15.)  cosh(.r — j)  =  cosh  ;ir  cosh/  — sinhjr  sinhj/. 


(16.)  coshi,r=i/. 


i+cosh^ 


2 

(17.)  sinhzi!  — sinh2/  =  2  cosh|  {ii -\-  v)?a\\\\\{u—v). 

(18.)         cosh  z^  +  cosh  z' =  2  cosh  ^  (z/ -}-?/)  cosh  |(«—z/). 
(19.)         cosh «  —  cosh z/ =  2  sinh^(«-f-z/)sinh|(«  — z/). 


CHAPTER  VII 
MISCELLANEOUS     EXERCISES 

RELATION  OF  FUNCTIONS 

77.  Prove  the  following : 

(I.)  cos-r  =  sinjr  cot  jr. 

(2.)  cscjr  tsinxz=secx. 

(3.)  (tan X -f- cot x)  sin x  cos x=i. 

(4.)  (secj  — tanj)  (sec7+ tanj)=  i. 

(5.)  (CSC^-  — COt^-)  (CSC£'+COt^')=  I, 

(6.)  cos'j-f(tan^  —  cot /)  sin_j/ cosy  =  sin'j/. 

(7.)  cos*^  —  sin*^  +  I  =  2  cos' a-. 

(8.)  (sinj/  —  cos j)"  =  1  —  2  sin/  cos_y. 

(9.)  sin'.r-}-cos'^  =  (sin  jr-f-cosx)  (I  —  sin  x  cos^). 

.  cot  ;r  4- tan  y 

(10.)  ; =cot.rtan  V. 

tanx-fcot/ 

(II.)  cos'^y  — sin''_y  =  2  cos'j/ — I. 

(12.)  I  —  tan*x  =  2  sec'jr  —  sec*.r. 

cos  X 

(U-)  -■ :^— =  tanjr. 

•^     sin.r  cot'^r 

(14.)  sec'_>' csc''/  =  tan'7  +  cot''/-|-2. 

(15.)  cot/— esc/ sec  J/ (I— 2  510'/)=:  tan/. 

,  . ,  (    \  N''     I  —  cos^ 

(16.)  I    .      — cot^)  =- 
Vsin^  /       I 


(I7-) 


+  cos^ 
secy         tan/  —  sin/ 


i-f-cos/  sin'/ 


(18.)  \-\ =  (sin^-f  cosjr)'. 

sec;r 

(I9-)  — \ sin'^  =  (cos;r  — sin;r)  (i-}- sin.r  cos^r), 

^  ,  sec*  X 

(20.)  (sin  JT  cos/+cos;r  sin/)'-|-(cos.r  cos  y-- sin.r  sin/)''  =  I 


MISCELLANEOUS  EXERCISES  79 

(21.)  {aco%x  —  b  €\nxf -{-{a  €\wx-\-b  CQ's.xy' =  a^-^-b"^. 

,      N  '  .4  tan'' y 

(22.)  ;~:w5--Tr::^:A2  =  i  + ; 


(cos'^j  —  s  i  n '  _j')^  (I  —  tan  "^  yf 

Find  an  angle  not  greater  than  90°  which  satisfies  each  of  the  fol- 
lowing equations: 

(23.)  4  cos  .r  =  3  sec  x. 
(24.)  sinj  =  cscj  —  f. 
(25.)  -\/2  sinx  —  tan.r  =  o, 

(26.)  2  cos.r —  Vs  cota'  =  o. 
(27.)  tanj  +  cot^  — 2  =  0. 
(28.)  2  sin-jK — 2  =  — -^^2  cos_y. 
(29.)  3  tan^x—  I  =4  sin^x. 
(30.)  cos^.r4-2  sin-x  —  f  sin.r  =  o. 
(31.)  csc.i'  =  f  tanx. 

(32.)  sec  .1- 4"  tan -t' =  ^  "V/S- 
(33.)  tan  .r  -f-  2  -y/3  cos  x  =  o. 
(34.)  3  sinx — 2  cos^x=o. 

Express  the  following  in  terms  of  the  functions  of  angles  less 
than  45°: 

(35.)  sin  92°. 
(36.)  cos  127°. 
'(37.)  tan  320°. 
(38.)  cot  350°. 
(39.)  sin  265°. 
(40.)  tan  171°. 

(41.)  Given  sin  .1=:^  and  x  in  quadrant  II;  find  all  the  other 
functions  of  x. 

(42.)  Given  cos.t-  =  —  |  and  x  in  quadrant  III;  find  all  the  other 
functions  of  x. 

(43.)  Given  tan.r=:f  and  x  in  quadrant  III;  find  all  the  other 
functions  of  x. 

(44.)  Given  cotx=: — |  and  x  in  quadrant  IV;  find  all  the  other 
functions  of  x. 


8o  PLANE    TRIGONOMETRY 

In  what  quadrants  must  the  angles  lie  which  satisfy  each  of  the 
following  equations: 
(45.)  sin  .r  cos  .r  =  ^  y'3. 
(46.)  sec  X  tan  ;r  =  2  -y/3. 
(47.)  tan^z-j-  ■Y/20  cos_y=  o. 
(48.)  cos  X  cot  x  =  ^. 

Find  all  the  values  of  _y  less  than  360°  which  will  satisfy  the  fol- 
lowing equations : 

(49.)  tan_y  +  2  sinj/  =  o. 

(50.)  (I  -|-tan;r)(i  —  2  sin;f)=o. 

(51.)  sin  JT  cos  jjr  (I  -)-  2  cos  x)  =  o. 

Prove  the  following: 
(52.)  cos  780°  =  ^. 

(53.)  sini485°  =  ^-v/2. 
(54.)  cos  2550°  =  i-/3. 
(55.)  sin  (—  3000°)  =  —  cos  30° 
(56.)  cos  1 300°  =  — cos  40°. 

(57.)  Find  the  value  of  a  sin  90°  +  ^  tano°-\-a  cos-lSo®. 
(58.)  Find  the  value  of  a  sin  30°  +  *^  tan  45°  4-'^  cos  60°  +  ^  tan  135° 
(59.)  Find  the  value  of  (a—d)  tan  225°-|-<J  cos  i8o°  —  a  sin  270°. 
(60.)  Find  the  value  of  (a  sin  45°-}-  6  cos  45°)  («  sin  1 35°  +  (^  sin  225°). 

RIGHT  TRIANGLES 

78,  In  the  following  problems  the  planes  on  which  distances  are  measured 
are  understood  to  be  horizontal  unless  otherwise  stated, 

(I.)  The  angle  of  elevation  of  the  top  of  the  tower  from  a  point 
1121  ft.  from  its  base  is  observed  to  be  150  17';  find  the  height  of 
the  tower. 

(2.)  A  tree,  77  ft.  high,  stands  on  the  bank  of  a  river;  at  a  point  on 
the  other  bank  just  opposite  the  tree  the  angle  of  elevation  of  the 
top  of  the  tree  is  found  to  be  5°  17'  37".  Find  the  breadth  of  the 
river. 


MISCELLANEOUS  EXERCISES  81 

(3.)  What  angle  will  a  ladder  42  ft.  long  make  with  the  ground  if  its 
foot  is  25  ft.  from  the  base  of  the  building  against  which  it  is  placed  ? 

(4.)  When  the  altitude  of  the  sun  is  33°  22',  what  is  the  height  of  a 
tree  which  casts  a  shadow  75  ft. .'' 

(5.)  Two  towns  are  3  miles  apart.  The  angle  of  depression  of  one, 
from  a  balloon  directly  above  the  other,  is  observed  to  be  8°  15'. 
How  high  is  the  balloon  .? 

(6.)  From  a  point  197  ft.  from  the  base  of  a  tower  the  angle  of  ele- 
vation was  found  to  be  46°  45'  54"  ;  find  the  height  of  the  tower. 

(7.)  A  man  5  ft.  10  in.  high  stands  at  a  distance  of  4  ft.  7  in.  from 
a  lamp-post,  and  casts  a  shadow  18  ft.  long;  find  the  height  of  the 
lamp-post. 

(8.)  The  shadow  of  a  building  101.3  ft.  high  is  found  to  be  131. 5 
ft.  long;  find  the  elevation  of  the  sun  at  that  time. 

(9.)  A  rope  112  ft.  long  is  attached  to  the  top  of  a  building  and 
reaches  the  ground,  making  an  angle  of  "j^^  20'  with  the  ground ; 
find  the  height  of  the  building. 

(10.)  A  house  is  130  ft.  above  the  water,  on  the  banks  of  a  river; 
from  a  point  just  opposite  on  the  other  bank  the  angle  of  elevation 
of  the  house  is  14°  30'  21".     Find  the  width  of  the  river. 

(II.)  From  the  top  of  a  headland,  1217.8  ft.  above  the  level  of  the 
sea,  the  angle  of  depression  of  a  dock  was  observed  to  be  10°  9'  13' ; 
find  the  distance  from  the  foot  of  the  headland  to  the  dock. 

(12.)  1 121. 5  ft.  from  the  base  of  a  tower  its  angle  of  elevation  is 
found  to  be  11°  3'  5  '';  find  the  height  of  the  tower. 

(13.)  One  bank  of  a  river  is  94.73  ft.  vertically  above  the  water,  and 
subtends  an  angle  of  10°  54'  13"  from  a  point  directly  opposite  at  the 
water's  edge;  find  the  width  of  the  river. 

(14.)  The  shadow  of  a  vertical  cliff  113  ft.  high  just  reaches  a  boat 
on  the  sea  93  ft.  from  its  base ;  find  the  altitude  of  the  sun. 

(15.)  A  rope,  38  ft.  long,  just  reached  the  ground  when  fastened  to 
the  top  of  a  tree  29  ft.  high.  What  angle  does  it  make  with  the 
ground  ? 

(16.)  A  tree  is  broken  by  the  wind.     Its  top  strikes  the  ground  15 

ft.  from  the  foot  of  the  tree,  and  makes  an  angle  of  42°  28'  with  the 

ground.     Find  the  height  of  the  tree  before  it  was  broken. 
6 


82  PLANE   TRIGONOMETRY 

(17.)  The  pole  of  a  circular  tent  is  18  ft.  high,  and  the  ropes  reach- 
ing from  its  top  to  stakes  in  the  ground  are  37  ft.  long;  find  the 
distance  from  the  foot  of  the  pole  to  one  of  the  stakes,  and  the  angle 
between  the  ground  and  the  ropes. 

(18.)  A  ship  is  sailing  southwest  at  the  rate  of  8  miles  an  hour. 
At  what  rate  is  it  moving  south  ? 

(19.)  A  building  is  121  ft.  high.  From  a  point  directly  across  the 
street  its  angle  of  elevation  is  65°  3'.     Find  the  width  of  the  street. 

(20.)  From  the  top  of  a  building  52  ft.  high  the  angle  of  elevation 
of  another  building  112  ft.  high  is  30°  12'.  How  far  are  the  buildings 
apart  ? 

(21.)  A  window  in  a  house  is  24  ft.  from  the  ground.  What  is  the 
inclination  of  a  ladder  placed  8  ft.  from  the  side  of  the  building  and 
reaching  the  window.? 

(22.)  Given  that  the  sun's  distance  from  the  earth  is  92,000,000 
miles,  and  its  apparent  semidiameter  is  16'  2";  find  its  diameter, 

(23.)  Given  that  the  radius  of  the  earth  is  3963  miles,  and  that  it 
subtends  an  angle  of  57'  2"  at  the  moon;  find  the  distance  of  the 
moon  from  the  earth. 

(24.)  Given  that  when  the  moon's  distance  from  the  earth  is  238885 
miles,  its  apparent  semidiameter  is  15' 34";  find  its  diameter  in  miles. 

(25.)  Given  that  the  radius  of  the  earth  is  3963  miles,  and  that  it 
subtends  an  angle  of  9"  at  the  sun ;  find  the  distance  of  the  sun 
from  the  earth. 

(26.)  A  light-house  is  57  ft.  high ;  the  angles  of  elevation  of  the  top 
and  bottom  of  it.  as  seen  from  a  ship,  are  5°  3'  20"  and  4°  28'  8".  Find 
the  distance  of  its  base  above  the  sea-level. 

(27,)  At  a  certain  point  the  angle  of  elevation  of  a  tower  was  ob- 
served to  be  53°  51'  16",  and  at  a  point  302  ft.  farther  away  in  the 
same  straight  line  it  was  9°  52'  10";  find  the  height  of  the  tower. 

(28.)  A  tree  stands  at  a  distance  from  a  straight  road  and  between 
two  mile-stones.  At  one  mile-stone  the  line  to  the  tree  is  observed 
to  make  an  angle  of  25°  15'  with  the  road,  and  at  the  other  an  angle 
of  45°  17'.     Find  the  distance  of  the  tree  from  the  road. 

(29.)  From  the  top  of  a  light-house,  225  ft.  above  the  level  of  the 
sea,  the  angle  of  depression  of  two  ships  are  17°  21'  50"  and  13°  50'  22", 


MISCELLANEOUS  EXERCISES  83 

and  the  line  joining  the  ships  passes  directly  beneath  the  light-house  ; 
find  the  distance  between  the  two  ships. 

ISOSCELES   TRIANGLES   AND    REGULAR   POLYGONS 

79.  (I.)  The  area  of  a  regular  dodecagon  is  37.52  ft. ;  find  its 
apothem. 

(2.)  The  perimeter  of  a  regular  polygon  of  11  sides  is  23.47  ft. ;  find 
the  radius  of  the  circumscribing  circle. 

(3.)  A  regular  decagon  is  circumscribed  about  a  circle  whose  radius 
is  3.147  ft. ;  find  its  perimeter. 

(4.)  The  side  of  a  regular  decagon  is  23.41  ft. ;  find  the  radius  of 
the  inscribed  circle. 

(5.)  The  perimeter  of  an  equilateral  triangle  is  17.2  ft.;  find  the 
area  of  the  inscribed  circle. 

(6.)  The  area  of  a  regular  octagon  is  2478  sq.  in. ;  find  its  pe- 
rimeter. 

(7.)  The  area  of  a  regular  pentagon  is  32.57  sq.  ft. ;  find  the  radius 
of  the  inscribed  circle. 

(8.)  The  angle  between  the  legs  of  a  pair  of  dividers  is  43°,  and  the 
legs  are  7  in.  long ;  find  the  distance  between  the  points. 

(9.)  A  building  is  37.54  ft.  wide,  and  the  slope  of  the  roof  is  43°  36' ; 
find  the  length  of  the  rafters. 

(10.)  The  radius  of  a  circle  is  12733,  and  the  length  of  a  chord  is 
18321 ;  find  the  angle  the  chord  subtends  at  the  centre. 

(II.)  If  the  radius  of  a  circle  is  taken  as  unity,  what  is  the  length 
of  a  chord  which  subtends  an  angle  of  'Ji°  17'  40".^ 

(12.)  What  angle  at  the  centre  of  a  circle  does  a  chord  which  is  ^ 
of  the  radius  subtend  .' 

(13.)  What  is  the  radius  of  a  circle  if  a  chord  11223  ft.  subtends  an 
angle  of  59°  50'  52" .' 

(14.)  Two  light-houses  at  the  mouth  of  a  harbor  are  each  2  miles 
from  the  wharf.  A  person  on  the  wharf  finds  the  angle  between  the 
lines  to  the  light-houses  to  be  17°  32'.  Find  the  distance  between  the 
two  light-houses. 

(15.)  The  side  of  a  regular  pentagon  is  2;  find  the  radius  of  the 
inscribed  circle. 


84  PLANE   TRIGONOMETRY 

(i6.)  The  perimeter  of  a  regular  heptagon  inscribed  in  a  circle  is 
12  ;  find  the  radius  of  the  circle. 

(17.)  The  radius  of  a  circle  inscribed  in  an  octagon  is  3;  find  the 
perimeter  of  the  octagon. 

(18.)  A  regular  polygon  of  9  sides  is  inscribed  in  a  circle  of  unit 
radius;  find  the  radius  of  the  inscribed  circle. 

(19.)  Find  the  perimeter  of  a  regular  decagon  circumscribed  about 
a  unit  circle. 

(20.)  Find  the  area  of  a  regular  hexagon  circumscribed  about  a 
unit  circle. 

(21.)  Find  the  perimeter  of  a  polygon  of  11  sides  inscribed  in  a 
unit  circle. 

(22.)  The  perimeter  of  a  dodecagon  is  30;  find  its  area. 

(23.)  The  area  of  a  regular  polygon  of  11  sides  is  18;  find  its  pe- 
rimeter. 

TRIGONOMETRIC  IDENTITIES  AND  EQUATIONS 
80,  Prove  the  following : 


(I.)  sin  ^j/±cos  |j/  =  -v/i  isinj/. 

,   .  cos;r  — cosv  ,  ,     ,     ^        -,  ,         v 

(2.)  ; ^-  =  —tan  4  {X  +  y)  tan  \  ix  —y). 

cosjr  +  cosj  «v     1.//        8v       J) 

,    ^  sin  IX  4-  sin4:r 

(3) -, -^  =  tan3jr. 

cos  2x  +  cos  i,x 

(4.)  cos-7tan'j  +  sin*jcot'_y=i. 

,    .    co?,{x  -V-y-\-z) 

(5.)  — —    .        .      =  cot  X  coty  cot^r  — cot;t-  — coty— cot^. 
sm.r  sm^  sm^r  •' 

(6.)  cos'  {x  — ;-)  —  sin"  (x  -\-y)  =  cos  2x  cos  2_y. 

,    .    sin-r  +  siny  ,  ^  . 

(7.)  ~ ^  =  —  cot  4  (.r  —  y). 

cos  JT-T-COS^  s\         y 

^^^  cosjr  — secjT  9,      ,      J 1  ,v 

(8) g -— = 4  cos' i X (cos' ^x  —  i). 

sin  2x 
(9.)  cot;r  =  - 


I  —  cos  2X 


,     V         Q         I  —  COS  2y 
(10.)  tan' y  =  — ; —  • 

^  -^         I  +  cos  2Jf 

(I  I .)  cot  X  —  tan  x=.2  cot  2x. 


(12. 

(13- 

(14. 
(15- 
(.6. 

(>7. 
(i8. 

(19- 

(20. 
(21. 

(22. 

(23- 

(24. 

(25- 
(26. 

(27. 

(28. 

(29. 
(30 

(31 

(32- 

(33- 
(34- 

(35 


MISCELLANEOUS  EXERCISES  85 

tan  I  ;r  +  2  sin'' ^  ;r  cot  ;ir  =  sin  jr. 

tan  X  ±  tan  y 

— ^=  ±sin;ir  sec;ir  tany. 

cot  X  ■±L  cot/ 

sin  X  —  2  sin'  x  =  sin  x  cos  2^. 

4  sin^y  sin  (60° — y)  sin  (60°  -j-j)  =  sin  3/. 

sin_>'(i  — tan''/) 


\cosy  —  siny      cosy+siny/ 


sec\y  \cos_y  —  sin/      cos/ +  sin// 

I  4-  tan/  tan  ^/  =:sec/. 

sin  4,r  =  4  sin  .r  cos'  x  —  4  cos;ir  sin' jr. 

2 

sec  2.r+  tan  2x4- 1  = • 

I  —  tan  X 

tan  50°  + cot  50°=:  2  sec  10°. 

cos  (x  +  45°)  +  sin  (x  —  45°)  =  o. 

tan  jr 


sm  2/. 


I  —  cot  2x  tan  X 


=:sin  2X. 


„    ,    .  .  /i  —  cos  2Jir 

(I  —  tan'' jr)  sin  x  cos  jr  =  cos  2x\/ — ; . 

*   I  -|-cos  2X 

cos/ +  sin/ 

— ~ .—--  =  tan  2/  + sec  2/. 

cos/  —  sm/ 

sin  (jr-f-/)  cos  jr — cos(,r+/)  sin  j-  =  sin/. 

cos  (x — j)  sin/ +  sin  {x—y)  cos/ =  sin  jr. 

sin  (JT— /)        sin  (/  — g)       sin  (.sr  — jr)  _ 

cosjr  cos/      cos/  cos.sr      cos  2"  cosjt 

sin  jr+sin  2.r 

■ =  cot  4  X. 

cos  X  —  cos  2X 

2  sin"  X  sin^/^-  2  cos"  jt  cos'/  =  i  +  cos  2jr  cos  2/. 

sin  60°  +  sin  30°  =  2  sin  45°  cos  1 5°. 

tan  (jr— /)  +  tan/   _^ 

I  —  tan  (-*■—./)  tan/ 

2 


:  tan  X. 


-— -i+cot"^/. 


sin/  tan  |/ 
sin  4Jtr+sin  2jrr=2  sin  ^x  cos  jr. 
sinjr-j-sin/  _ cos .r -}- cos/ 
cosj:  —  cos/       sin/  —  sinjr 


2-V/2 
(36.)  2  tan  2/  =  tan(45°+/)  — tan(45°— /). 


86 


PLANE    TRIGONOMETRY 


tan  2 ^  +  tan x _%\x\yc 
tan2jr  —  tanjr       sin;r 

3  tany  —  tan'y 

tan  31/  = -2 ^ — ^. 

1—3  tan"/ 

sin 60° -f-  sin  20°  =  2  sin  40°  cos  20°. 

sin 40°  —  sin  10°  =  2  cos 25°  sin  15°. 

cos 2^ — cos4-r  =  2  sin3-r  sinjT. 

tan  1 50  =  2  —  v/3. 

("v/i  +sin^  — Vi  —%\nxY-=\  sin'i^;*-. 

(-y/i  4-sin;jr  +  v  I  —  sinj)''  =  4  cos*^;r. 

sin  (2^  +  1/)  ,     .     .       sin/ 
^ — -^  —  2  cos(x -\-y)  =  -^-^ . 


sin^r 

sin4jr 

.  =  2  cos  IX. 

sin2  jr 

sin  50°  —  sin  70° -f- sin  10°  =  0. 

TV  TT  .       5""     .       TT 

cos cos  -  =  2sin;:^sin  — 

32  12       12 


I  —  tan''(45°  — ;ir) 
I +"1311^(45°^  x) 
si 


=  sin2jr. 


5in75°  — sin  15°  _      H 
:os75°  +  cos  1 5° ~  V  V 


tan  H  ^(  I  +  cot"  \xf=  -^^— -• 

tan  750  =  2 +  -\/3- 

sin  3J'4-sin  5-r  =  2  sin4Jr  cos;t:. 

cos  5^  -\-  cos  9^:  =  2  cos  "jx  cos  2X. 

Sin  15°=  ^  -^  .,■ 


sin3x 


2\/2 

sinjf 


=  tan  X. 


cos3.r  +  cosj: 

sin  5J=  5  sinj— 20  siny+  16  sin'j/. 

cos  SJ'  =  S  cos^ —  20  cos'j  +  '6  cos^. 

4tan;r(i — tan'.r) 
^'"^-^=   (i+tan'-r)-   " 
cos(45°-f-jr)-|-(cos450  — a)=:  -\/2  cos;r. 
cos  3a-  4-  cos  5^  -H  cos  7x  +  cos  1 5;r  =  4  cos  4x  cos  5  ;r  cos  6x. 


MISCELLANEOUS  EXERCISES  87 

(62.)  ixv? \x (cot\x — if  =  1— sin.r. 

,,    ^    3sin,r  — sin3.r 

(63.)  ^ ^     =tan^r. 

cos  3-^  4"  3  cos  X 

(64.)  sinx([  -j- tanx)4-cos.r(i  +cot.i-)  =  csc.r  +  sec;r. 

,,    ,  cosV  —  sin^jr      24-sin2.r 

65.)  ^    =_r 

cosx  —  sinjr  2 

(66.)  cos_y  +  cos  (120— _y)  +  cos(i20  ■\-y)  =  o. 

, ,    ,  sin  3JI" 

(67.)  —. =  2  cos 2^+ I. 

sinx 

(68)   (<^°sj^~  cos  3/)(sin  8_>/  +  sin  27)  _  ^ 
(sin  5y  —  sin_y)(cos4>' —  cos6_>') 


(69.)  ( ^'""  y=-^- 

^    ^^    Vl+COSX/  I 


+  cosx/       I  +cos;r 

.    sin3.r      cos3-r 
(70.)  -^~-—^-  =2. 
sin-r        cos^ 

.    I  4-sin^ -|-cos.r 

(7I-)  --f- ^ =cotijr. 

i  +  sif^  —  cos^ 

sin(4^-2^)-|-sin(4^-2.)^^^ 

cos  (4Jir  —  2_>')  +  cos  (4j/  —  2;r) 

^     sin^r  +  sin  3.r  +  sin  5.r  +  sin  7.r 

A  (73-)  T- ^—7- T  =  tan  4.r. 

cos  .r  4-  cos  3.r  +  cos  5.r  +  cos  7-r 

If  A,  B,  and  C  are  the  angles  of  a  triangle,  prove  the  following  ; 
(74.)  sin  2A  -j-  sin  2B  -\-  sin  2(7  =  4  sin  A  sin  B  sin  C. 
(75.)  sin  2.^  +  sin2^  —  sin  2C  =  4  cos^  cosi?  sin  C 
(76.)  sin''/i4-sin''i54-sin''C=2  +  2  cos^  cos^cosC 
{J7.)  tan  ^  +  tan  j9  +  tan  C  =  tan  A  tan  B  tan  C. 

Solve  the  following  equations  for  values  of  x  less  than  360° 
(78.)  cos2x+cos.r  =  —  I. 
(79.)  sin  ,r-|-sin7;ir  =  sin4_r. 
(80.)  cos;r  —  sin2jr  —  cos3;r  =  o. 
'(81.)  cosjr  —  sin  3;tr  —  cos2.r  =  o. 
(82.)  sin4jr — 2  sin2;ir=r:o. 
(83.)  sin  2;r  —  cos2x  —  sinx  +  cos.r  =  0. 
(84.)  sin  (60°  —  ;r)  —  sin  (60°  +  ,t)  =  +  ^  ^ J^ 
(85.)  sin(30°  +  ;ir)  — cos(6o°  +  x)  =  — ^^3, 


88  PLANE    TRIGONOMETRY 

\  (86.)  csc;r=  i  -f-cot-r. 
(87.)  cos2;ir  =  cos'-r. 
(88.)  2  sinj=sin2_y. 
(89.)  sin  3/ +  sin  2/  + sin  J  =  0. 
(90.)  sin'jr-|-  5  cos'^r  =  3. 
(91.)  tan(45°  — jr)  +  cot(45^-.r)=4. 

OBLIQUE  TRIANGLES 

81.  (I.)  It  is  required  to  find  the  distance  between  two  points,  A 
and  B,  on  opposite  sides  of  a  river.  A  line,  /?C,  and  the  angles  BAC 
and  ACB  are  measured  and  found  to  be  2483  ft.,  61°  25',  and  52°  17' 
respectively. 

(2.)  A  straight  road  leads  from  a  town  .^  to  a  town  B,  12  miles 
distant ;  another  road,  making  an  angle  of  TJ°  with  the  first,  goes  from 
A\.o  ^  town  C,  7  miles  distant.     How  far  are  the  towns  B  and  C apart  .^ 

(3.)  In  order  to  determine  the  distance  of  a  fort.  A,  from  a  battery, 
B,  a  line,  BC,  one-half  mile  long,  is  measured,  and  the  angles  ABC 
and  ACB  are  observed  to  be  75°  18'  and  78°  21'  respectively.  Find 
the  distance  ^^. 

(4.)  Two  houses,  A  and  B,  are  1728  ft.  apart.  Find  the  distance  of 
a  third  house,  C,  from  A  if  BAC—M°  51' and  ABC=  $7°  23'. 

(5.)  In  order  to  determine  the  distance  of  a  bluff,  A,  from  a  house, 
B,  in  a  plane,  a  line,  BC,  was  measured  and  found  to  be  1281  yards, 
also  the  angles  ABC  and  BCA  65°  31'  and  70°  2'  respectively.  Find 
the  distance  AB. 

(6.)  Two  towns,  3  miles  apart,  are  on  opposite  sides  of  a  balloon. 
The  angles  of  elevation  of  the  balloon  are  found  to  be  13°  19'  and 
20°  3'.     Find  the  distance  of  the  balloon  from  the  nearer  town, 

(7.)  It  is  required  to  find  the  distance  between  two  posts,  .(4  and  B, 
which  are  separated  by  a  swamp.  A  point  C  is  1272.5  ft.  from  A.  and 
2012.4  ft-  from  B.     The  a.ng\e  ACB  is  41°  9'  11". 

(8.)  Two  stakes,  A  and  B,  are  on  opposite  sides  of  a  stream  ;  a 
third  point,  C,  is  so  situated  that  the  distances  AC  and  BC  can  be 
found,  and  are  431.27  yards  and  601.72  yards  respectively.  The  angle 
ACB  is  39°  53'  13".     Find  the  distance  between  the  stakes  A  and  B. 


MISCELLANEOUS   EXERCISES  89! 

(9.)  Two  light-houses,^  and  B,  are  11  miles  apart.  A  ship,  C,  is 
observed  from  them  to  mai<e  the  angles  BAC=^2^\°  13'  31"  and  ABC 
=z2i°  46'  8".     Find  the  distance  of  the  ship  from  A. 

(10.)  Two  islands,  A  and  B,  are  6103  ft.  apart.  Find  the  distance 
from  A  to  a  ship,  C,  if  the  angle  ABC  is  37°  25'  and  BAC  is  40°  32'. 

(II.)  In  ascending  a  cliff  towards  a  light-house  at  its  summit,  the 
light-house  subtends  at  one  point  an  angle  of  21°  22'.  At  a  point 
55  ft.  farther  up  it  subtends  an  angle  of  40°  27'.  If  the  light-house 
is  58  ft.  high,  how  far  is  this  last  point  from  its  foot.'' 

(12.)  The  distances  of  two  islands  from  a  buoy  are  3  and  4  miles 
respectively.  The  islands  are  2  miles  apart.  Find  the  angle  sub- 
tended by  the  islands  at  the  buoy. 

(13.)  The  sides  of  a  triangle  are  151.45,  191.32,  and  250.91.  Find 
the  length  of  the  perpendicular  from  the  largest  angle  upon  the 
opposite  side. 

(14.)  A  tree  stands  on  a  hill,  and  the  angle  between  the  slope  of  the 
hill  and  the  tree  js  110°  23'.  At  a  point  85.6  ft.  down  the  hill  the 
tree  subtends  an  angle  of  22°  22'.     Find  the  height  of  the  tree. 

(15.;  A  light-house  54  ft.  high  is  built  upon  a  rock.  From  the  top 
of  the  light-house  the  angle  of  depression  of  a  boat  is  19°  10',  and 
from  its  base  the  angle  of  depression  of  the  boat  is  12°  22'.  Find  the 
height  of  the  rock  on  which  the  light-house  stands. 

(16.)  Three  towns,  A,  B,  and  C,  are  connected  by  straight  roads. 
AB-=.^  miles,  BC=  5  miles,  and  AC=y  miles.  Find  the  angle  made 
by  the  roads  AB  and  BC. 

(17.)  Two  buoys,  A  and  B,  are  one-half  mile  apart.  Find  the  dis- 
tance from  A  to  &  point  C  on  the  shore  if  the  angles  ABC  and  BAC 
are  77°  7'  and  67°  17'  respectively. 

(18.)  The  top  of  a  tower  is  175  ft.  above  the  level  of  a  bay.  From 
its  top  the  angles  of  depression  of  the  shores  of  the  bay  in  a  certain 
direction  are  57°  16'  and  15°  2'.     Find  the  distance  across  the  bay. 

(19.)  The  lengths  of  two  sides  of  a  triangle  are  \/2  and  y'3.  The 
angle  between  them  is  45°.     Find  the  remaining  side. 

(20.)  The  sides  of  a  parallelogram  are  172.43  and  101.31,  and  the 
angle  included  by  them  is  61°  16'.     Find  the  two  diagonals. 

(21.)  A  tree  41  ft.  high  stands  at  the  top  of  a  hill  which  slopes 


90  PLANE    TRIGONOMETRY 

io°  12'  to  the  horizontal.  At  a  certain  point  down  the  hill  the  tree 
subtends  an  angle  of  28°  29'.  Find  the  distance  from  this  point  to 
the  toot  of  the  tree. 

(22.)  A  plane  is  inclined  to  the  horizontal  at  an  angle  of  7°  33'.  At 
a  certain  point  on  the  plane  a  flag-pole  subtends  an  angle  20°  3',  and  at 
a  point  50  ft.  nearer  the  pole  an  angle  of  40°  35'.  Find  the  height  of 
the  pole. 

(23.)  The  angle  of  elevation  of  an  inaccessible  tower,  situated  in  a 
plane,  is  53°  19'.  At  a  point  227  ft.  farther  from  the  tower  the  angle 
of  elevation  is  22°  41'.     Find  the  height  of  the  tower. 

(24.)  A  house  stands  on  a  hill  which  slopes  12°  18'  to  the  horizontal. 
75  ft.  from  the  house  down  the  hill  the  house  subtends  an  angle  of 
32°  5'.     Find  the  height  of  the  house. 

(25.)  From  one  bank  of  a  river  the  angle  of  elevation  of  a  tree  on 
tiie  opposite  bank  is  28°  31'.  From  a  point  139.4  ft.  farther  away  in  a 
direct  line  its  angle  of  elevation  is  19°  10'.     Find  the  width  of  the  river. 

(26.)  From  the  foot  of  a  hill  in  a  plane  the  angle  of  elevation  of 
the  top  of  the  hill  is  21°  7'.  After  going  directly  away  211  ft.  farther, 
the  angle  of  elevation  is  18°  37'.     Find  the  height  of  the  hill. 

(27.)  A  monument  at  the  top  of  a  hill  is  153.2  ft,  high.  At  a  point 
321.4  ft.  down  the  hill  the  monument  subtends  an  angle  of  11°  13'. 
Find  the  distance  from  this  point  to  the  top  of  the  monument. 

(28.)  A  building  is  situated  on  the  top  of  a  hill  which  is  inclined 
10°  12'  to  the  horizontal.  At  a  certain  distance  up  the  hill  the  angle 
of  elevation  of  the  top  of  the  building  is  20°  55',  and  11 5.3  ft.  farther 
down  the  hill  the  angle  of  elevation  is  15°  10'.  Find  the  height  of 
the  building. 

(29.)  A  cloud,  C,  is  observed  from  two  points,  A  and  B,  2874  ft. 
apart,  the  line  AB  being  directly  beneath  the  cloud.  At  A,  the  angle 
of  elevation  of  the  cloud  is  T]°  19',  and  the  angle  CAB  is  51°  18'. 
The  angle  ABC  is  found  to  be  60°  45'.  Find  the  height  of  the  cloud 
above  A. 

(30.)  Two  observers.  A  and  B,  are  on  a  straight  road,  675.4  ft.  apart, 
directly  beneath  a  balloon,  C  The  angles  ABC  and  BAC  are  34°  42' 
and  41°  15'  respectively.  Find  the  distance  of  the  balloon  from  the 
first  observer. 


MISCELLANEOUS  EXERCISES  91 

(31.)  A  man  on  the  opposite  side  of  a  river  from  two  objects,  A 
and  B,  wishes  to  obtain  their  distance  apart.  He  measures  the  dis- 
tance CD  —  i^-]  ft.,  and  the  angles  ACB=2g°  33',  BCD  =  38°  52'.  ADB 
=  54°  10',  and  ^Z>C=34°  11'.     Find  the  distance  AB. 

(32.)  A  cliflf  is  327  ft.  above  the  sea-level.  From  the  top  of  the 
cliff  the  angles  of  depression  of  two  ships  are  15°  11'  and  13°  13'. 
From  the  bottom  of  the  cliff  the  angle  subtended  by  the  ships  are 
122°  39'.     How  far  are  the  ships  apart } 

(33.)  A  man  standing  on  an  inclined  plane  112  ft.  from  the  bottom 
observed  the  angle  subtended  by  a  building  at  the  bottom  to  be  33° 
52'.  The  inclination  of  the  plane  to  the  horizontal  is  18°  51'.  Find 
the  height  of  the  building. 

(34)  Two  boats,  A  and  B,  are  451.35  ft.  apart.  The  angle  of  ele- 
vation of  the  top  of  a  light-house,  as  observed  from  A,  is  33°  17'. 
The  base  of  the  light-house,  C,  is  level  with  the  water;  the  angles 
ABC  Sin6  CAB  are  12°  31'  and  137°  22'  respectively.  Find  the  height 
of  the  light-house. 

(35.)  From  a  window  directly  opposite  the  bottom  of  a  steeple  the 
angle  of  elevation  of  the  top  of  the  steeple  is  29°  21'.  From  another 
window,  20  ft.  vertically  below  the  first,  the  angle  of  elevation  is  39°  3'. 
Find  the  height  of  the  steeple. 

(36.)  A  dock  is  I  mile  from  one  end  of  a  breakwater,  and  i^  miles 
from  the  other  end.  At  the  dock  the  breakwater  subtends  an  angle 
of  31°  11'.     Find  the  length  of  the  breakwater  in  feet. 

(37.)  A  straight  road  ascending  a  hill  is  1022  ft.  long.  The  hill 
rises  i  ft.  in  every  4.  A  tower  at  the  top  of  the  hill  subtends  an 
angle  of  7°  19'  at  the  bottom.     Find  the  height  of  the  tower. 

(38.)  A  tower,  192  ft.  high,  rises  vertically  from  one  corner  of  a 
triangular  yard.  From  its  top  the  angles  of  depression  of  the  other 
corners  are  58°  4'  and  17°  49'.  The  side  opposite  the  tower  subtends 
from  the  top  of  the  tower  an  angle  of  75°  15'.  Find  the  length  of 
this  side. 

(39.)  There  are  two  columns  left  standing  upright  in  a  certain  ruins ; 
the  one  is  66  ft.  above  the  plain,  and  the  other  48.  In  a  straight  line 
between  them  stands  an  ancient  statue,  the  head  of  which  is  100  ft. 
from  the  summit  of  the  higher,  and  84  ft.  from  the  top  of  the  lower 


92  PLANE    TRIGONOMETRY 

column,  the  base  of  which  measures  just  74  ft.  to  the  centre  of  the 
figure's  base.  Required  the  distance  between  the  tops  of  the  two 
columns. 

(40.)  Two  sides  of  a  triangle  are  in  the  ratio  of  1 1  to  9,  and  the 
opposite  angles  have  the  ratio  of  3  to  i.     What  are  these  angles  } 

(41.)  The  diagonals  of  a  parallelogram  are  12432  and  8413,  and  the 
angle  between  them  is  78°  44' ;  find  its  area. 

(42.)  One  side  of  a  triangle  is  1 01 2.6  and  two  angles  are  52°  21'  and 
570  32' ;  find  its  area. 

(43.)  Two  sides  of  a  triangle  are  218.12  and  123.72,  and  the  included 
angle  is  59°  10' ;  find  its  area. 

(44.)  Two  angles  of  a  triangle  are  35°  15'  and  47°  18',  and  one  side 
is  2104.7  ;  find  its  area. 

(45.)  The  three  sides  of  a  triangle  are  1.2371,  1.4713,  and  2.0721  ; 
find  the  area. 

(46.)  Two  sides  of  a  triangle  are  168.12  and  179.21,  and  the  included 
angle  is  41°  14' ;  find  its  area. 

(47.)  The  three  sides  of  a  triangle  are  51  ft.,  48.12  ft.,  and  32.2  ft. ; 
find  the  area. 

(48.)  Two  sides  of  atriangle  are  in. 18  and  121. 21,  and  the  included 
angle  is  27°  50' ;  find  its  area. 

(49.)  The  diagonals  of  a  parallelogram  are  37  and  51,  and  they  form 
an  angle  of  65° ;  find  its  area. 

(50.)  If  the  diagonals  of  a  quadrilateral  are  34  and  56,  and  if  they 
intersect  at  an  angle  of  67°,  what  is  the  area? 


SPHERICAL  TRIGONOiMETRY 

CHAPTER   VIII 

RIGHT  AND   QUADRANTAL  TRIANGLES 

RIGHT   TRIANGLES 

82,  Let  O  be  the  centre  of  a  sphere  of  unit  radius,  and 
ABC  a  right  spherical  triangle,  right  angled  at  A,  formed  by 
the  intersection  of  the  three  planes  A  OC,  A  OB,  and  BOC 


with  the  surface  of  the  sphere.  Suppose  the  planes  DAC" 
and  BEC  passed  through  the  points  A  and  B  respectively, 
and  perpendicular  to  the  line  OC.  The  plane  angles  DC" A 
and  BC'E  each  measure  the  angle  C  of  the  spherical  tri- 
angle, and  the  sides  of  the  spherical  triangle  «,  b,  c  have  the 
same  numerical  measure  as  BOC,  AOC,  and  A  OB  respec- 


94  SPHERICAL    TRIGONOMETRY 

tivcly,  then,  AD  =  \.z.wc,  BE  — s\n€,  BC  =  sina,  OC  =  cosa, 
0C"  =  cos d,  OB  =  cose,  AC"  =  sm  b. 

In  the  two  similar  triangles  OEC  and  OAC" , 
cos  c     cos  c      cos  a 


OA  1         cos  b 

In  the  triangle  BC'E, 


,  or  cos «  =  cos ^  cost:.  (i) 


•    ^     BE  .    ^     s\x\c  I-. 

sinC  =  -7r7T-,,or  sm6= -r •  (2) 

BC  s\w  a 


In  the  triangle  DAC'\ 


^      DA          ,       ^     tan  c  ,  ^ 

tan  C=^:rr5,  or  tan  C=^j^.  (3) 

Combining  formulas  (2)  and  (3)  with  (i), 

^     tan  b  f  . 

cos  6  = (4) 

tan  a  ^  ' 

Again,  if  AB  were  made  the  base  of  the  right  spherical 
triangle  ABC,  we  should  have 

sinB=- (5) 

sma 

„     tan  b  /^x 

tan^=-^ (6) 

sm  c 

cosB= (7) 

tanrt:  ^ 

From  the  foregoing  equations  we  may  also  obtain   by 

combinations, 

cos5=sin  C  cos<^.  (8) 

cos^"— sin  ^  cose.  (9) 

cosrt=cot^  cot  C  (10) 

NAPIER'S  RULES  OF  CIRCULAR  PARTS 

S3,  The  above  ten   formulas  are  sufficient  to  solve  all 
cases  of  right  spherical  triangles.     They  may,  however,  be 


RIGHT  AND   QUADRANTAL    TRIANGLES  95 

expressed  as  two  simple   rules,  called,  after  their  inventor, 
Napier's  rules. 

The  two  sides  adjacent  to  the  right  angle,  the  complement 
of  the  hypotenuse,  and  the  complements  of  the  oblique  an- 
gles are  called  the  circular  parts. 

'I'lie  right  angle  is  not  one  of  the  circular  parts. 


comp  B 


comp  a 


comp  (T- 


Thus  there  are  five  circular  parts — namely,  /^  c,  comprt,  compi5,  compC. 
Any  one  of  tlie  five  parts  may  be  called  the  middle  part,  then  the  two  parts  next 
to  it  are  called  adjacent  parts,  and  the  remaining  two  parts  are  called  the  opl'o- 
site  parts. 

Thus  if  c  is  taken  for  the  middle  part,  comp  B  and  b  are  adjacent  parts,  and 
comp  a  and  comp  Care  opposite  parts. 

The  ten  formulas  may  be  written  and  grouped  as  follows  : 


\st  Group. 
sin  comp  C  =  tan  comprt  tan  b. 
sin  comp^=tan  comp^  tan  c. 
sin  comp  a  =tan  comp^  tan  comp  C. 
sin  r  =:tan  compi9  tan^. 

sin  b  =tan  coiTip  C  tan  c. 


'id  Group. 
sin  comp rt= cos/' cose, 
sin  (5= cos  comp  (7  cos  comp^. 

sin  rrrcos  comprt  cos  comp  C. 

sin  compi9=cos  comp  C  cos  b. 
sin  comp  C=cos  comp^  cosf. 


Napier's  rules  may  be  stated  : 

I.  The  sine  of  the  middle  part  is  equal  to  the  product  of 
the  tangents  of  the  adjacent  parts. 

II.  The  sine  of  the  middle  part  is  equal  to  the  product  of 
the  cosines  of  the  opposite  parts. 


96 


SPHERICAL   TRIGONOMETRY 


84,  In  the  right  spherical  triangles  considered  in  this  work,  each 
side  is  taken  less  than  a  semicircumference,  and  each  angle  less  than 
two  right  angles. 

In  the  solution  of  the  triangles,  it  is  to  be  observed, 

(i.)  If  the  two  sides  about  the  right  angle  are  both  less  or  both 
greater  than  90°,  the  hypotenuse  is  less  than  90°;  if  one  side  is  less 
and  the  other  greater  than  90°,  the  hypotenuse  is  greater  than  90°. 

(2.)  An  angle  and  the  side  opposite  are  either  both  less  or  both 
greater  than  90°. 

EXAMPLE 


85.  Given  a  =  63°  56',  ^  =  40°  o',  to  find  c,  B,  and  C 


7V?  find  c. 
comp  a  is  the  middle  part. 
c  and  b  are  the  opposite  parts, 
sin  comp  rt = cos  ^  cosr, 
cosrt=cos  b  cos  c. 

COSrt 

cos  f  = -• 

cos  0 

log  cos  rt  =  9. 64288 

colog  cos  ^=0.11575 

log  cos  f =9.75863 

f=54°59'  47" 


To  find  C. 

comp  C  is  the  middle  part. 

comp  a,  and  b  are  adjacent  parts. 

sin  comp  C=tan  comprt  tan  b^ 

cos  C=cot  rt  tan  b. 


log  cot  rt=9  68946 
log  tan  ^=9  92381 
9  61327 
C=65°  45'  58" 


sin  B-- 


To  find  B. 
b  is  the  middle  part, 
comp  (/  and  comp  B  are  the  opposite 
parts, 
sin  ^=cos  comp  a  cos  comp  B, 
or  sin  /^=sin  a  sin  B. 

sin  b 
sinrt 

log  sin /^=9. 80807 

colog  sin  fl=o. 04659 

log  sin  .5=9.85466 

i?  =  45°4l'28" 

Check. 

Use  the  three  parts  originally  required. 

comp  C  is  the  middle  part. 

comp  .5  and  c  are  opposite  parts. 

sin  comp  C=cosr  cos  comp  B, 

or  cos  C=cos  c  %\n  B. 

log  cos  f =9. 75863 
log  sin  Z^=9. 85466 
log  cos  C=9.6i329 

C=65°  45'  54" 


RIGHT  AND  QUADRANTAL    TRIANGLES 


97 


AMBIGUOUS  CASE 

86.  When  a  side  about  the  right  angle  and  the  angle  opposite 
this  side  are  given,  there  are  two  solutions,  as  illustrated  by  the  fol- 
lowing figure.  Since  the  solution  gives  the  values  of  each  part  in 
terms  of  the  sine,  the  results  are  not  only  the  values  of  a,  b,  B,  but 
i8o°— rt,  \^o°—b,  iSo°~B. 


Given  r  =  26°  4'. 
C=36°o'. 


To  find  a,  a',  b,  b'  and  B,  B',  usin 
To  find  B  and  B' . 

sin  comp  C=  cos  comp  B  cose, 

or  cos  C^sin  B  cos  c, 

.     „     cos  C 

or  sin/>= • 

cos  c 

log  cos  C=g.  90796 

colog  cos  r=o  04659 

log  sin ^  =  9.95455 

B=  64°  14'  30" 

^'  =  i8o°-i5=ii5°  45'  30" 

To  find  b  and  b' . 

sin  bz=.\.iXi  c  tan  comp  C, 

or  sin  ^=tan  c  cot  C. 

log  tan  ^=9.68946 

log  cot  C= 0.13874 

log  sin  ^=9. 82820 

b=  42°  19'  17" 
^'  =  180°— (J=I37°  40'  43" 


g  Napier's  rules. 

To  find  a  and  a'. 

sin  c=cos  comp  a  cos  Comp  C, 
or         sin  <-=sin  a  sin  C, 

sin  c 
or         sin  a=  -: — -  • 
sin  C 

log  sin  c=()  64288 

colog  sin  C=o.  23078 

log  sin  a=9. 87366 

a=  48°  22'  55"- 
a'  =  i8o<='— a=i3i°  37'  5"  + 
(Discrepancy  due  to  omitted  decimals.) 

Check. 
sin  ^=:cos  comp  a  cos  comp  />', 
sin  ^=sinfl  s\nB. 
log  am  a  or  ^'=9. 87366 
log  sin.fi  ori5'=9.95455 
log  sin  ^=9.82821 

bz=  42°  19'  21" 
^'=180°— 3=137°  40'  39" 


98  SPHERICAL    TRIGONOMETRY 

QUADRANTAL  TRIANGLES 
87*  Def. — A  quadrantal  triangle  is  a  spherical  triangle 
one  side  of  which  is  a  quadrant. 

A  quadrantal  triangle  may  be  solved  by  Napier's  rules  for 
right  spherical  triangles  as  follows : 

By  making  use  of  the  polar  triangle  where 

^  =  i8o°— fl'  a  =  i8o°  — ^' 

B=\%o°—h'  d=i8o°—B' 

C=i8o°  —  c'  /=i8o=— C 

we  see  that  the  polar  triangle  of  the  quadrantal  triangle  is 
a  right  triangle  which  can  be  solved  by  Napier's  rules. 
Whence  we  may  at  once  derive  the  required  parts  of  the 
quadrantal  triangle. 

EXAMPLE 

Given       A  =  136°  4'.             B  =  140°  o'.  a  =  90°  o'. 
The  corresponding  parts  of  the  polar  triangle  are 

rt'  =  63°56',            <5'  =  4o°o',  ^'  =  90°. 
By  Napier's  rules  we  find 

^■  =  450  41' 28",            C'  =  65°45' 58".  f=  54°  59' 47": 

whence,  by  applying  to  these  parts  the  rule  of  polar  triangles,  we 
obtain 

(5=  1340  18' 32".            c— 114°  1^'  2".  C=i25°o'i3". 

EXERCISES 

88.  (I.)  In  the  right-angled  spherical  triangle  ABC,  the  side  a= 
63°  56',  and  the  side  ^  =  40°.  Required  the  other  side,  c,  and  the 
angles  B  and  C. 

(2.)  In  a  right-angled  triangle  ABC,  the  hypotenuse  <?  =  91°  42',  and 
the  angle  B  =  g^°  6'.     Required  the  remaining  parts. 

(3.)  In  the  right-angled  triangle  ABC,  the  side  (^  =  26°  4',  and  the 
angle  ^  =  36°.     Required  the  remaining  parts. 

(4.)  In  the  right-angled  spherical  triangle  ABC,  the  side  c=$4°  30', 
and  the  angle  B=:44°  50'.     Required  the  remaining  parts. 

Why  is  not  the  result  ambiguous  in  this  case? 


RIGHT  AND   QUADRANTAL    TRIANGLES  99 

(5.)  In  the  right-angled  spherical  triangle  ABC,  the  side  <^  =  55^  28', 
and  the  side  ^  =  63°  15'.     Required  the  remaining  parts. 

(6.)  In  the  right-angled  spherical  triangle  ABC,  the  angle  Bz=6()° 
20',  and  the  angle  C  =  58°  16'.     Required  the  remaining  parts. 

(7.)  In  the  spherical  triangle  ABC,  the  side  a  =  go°,  the  angle  C= 
42°  10',  and  the  angle  A=.iis°  20'.     Required  the  remaining  parts. 
Hint. — The  angle  A  of  the  polar  triangle  is  a  right  angle. 

(8.)  In  the  spherical  triangle  ABC,  the  side  b  =  go°,  the  angle  C= 
69°  13'  46",  and  the  angle  ^^  =  72°  12'  4",  Required  the  remaining 
parts. 

(9.)  In  the  right-angled  spherical  triangle  yi^C,  the  angle  ^=23° 
27'  42",  and  the  side  b=.  loP  39'  40".  Required  the  angle  B  and  the 
sides  a  and  c. 

(10.)  In  the  right  spherical  triangle  y?^C,  the  angle  B=.^y°  54'  20", 
and  the  angle  C=6i°  50'  29".     Required  the  sides. 


CHAPTER   IX 
OBLIQUE-ANGLED  TRIANGLES 

89,  Let  O  be  the  centre  of  a  sphere  of  unit  radius,  and 
ABC  an  oblique-angled  spherical  triangle  formed  by  the 
three  planes  A  OB,  BOC,  and  AOC.     Suppose   the   plane 


AED  passed  through  the  point  A  perpendicular  to  AO,  in- 
tersecting the  planes  AOB,  BOC,  and  AOC,  in  AE,  ED, 
and  AD  respectively.  Then  AD = tan  b,  AE=tan  c,  OD— 
sec  b,  OE  =  sQCc. 

In  the  triangle  EOD, 

ED^  =  sec'/5  +  secV  —  2  sec^  seer  cos«. 
In  the  triangle  AED, 

ED^  =  tan«^  -f-  tan V  —  2  tan  d  tan  c  cos  A. 
Subtracting  these  two  equations  and  remembering  that 
sec''*^  — tan''(J=r,  we  have 
0  =  2  —  2  sec  i  sec  c  cos  a -{-2  tan  d  tan  c  cos  A. 
Reducing,  we  have 

cos a=:cos &  cosc+slnfr  sine  co8^.  (i) 


OBLIQUE-ANGLED    TRIANGLES  loi 

If  we  make  b  and  c  in  turn  the  base  of  the  triangle,  we  obtain  in  a 
similar  way, 

cos  b  ■=  cos  c  cos  a  +  sin  c  s\na  cos  B, 
and  cosc=^cosa  cosb-\-s\na  sxnb  cosC. 

Remark. — In  this  group  of  formulas  the  second  may  be  obtained 
from  the  first,  and  the  third  from  the  second,  by  advancing  one  letter 
in  the  cycle  as  shown  in  the  figure;  thus, writing  b  for 
a,  c  for  b,  a  for  c,  B  for  A,  C  for  B,  and  A  for  C.  The 
same  principle  will  apply  in  all  the  formulas  of  Oblique- 
Angled  Spherical  Triangles,  and  only  the  first  one  of 
each  group  will  be  given  in  the  text. 

00,  By  making  use  of  the  polar  triangle  where 
b=\?,o°-B'  B=iSo°-d' 

we  may  obtain  a  second  group  of  formulas. 

Substituting  these  values  of  a,  b,  c,  and  A  in  (i),  and  remembering 
that  cos(i8o''— ^)  =  — cos^  and  sin  (180°— /?)  =  sin^,  we  have 
cos^'  =  —  cosB'  cosC  +  sin^'  sin  C  cosa'. 
Since  this  is  true  for  any  triangle,  we  may  omit  the  accents  and 
write, 

cos A= —cosB  cos C+»inB  sinC cosa,  (2) 

FORMULAS   FOR    LOGARITHMIC   COMPUTATION 
91»  Formula  (i),  cosrt:  =  cos<5  cos^  +  sin  ^  sin  c  cosvi, 
cosa — cos/^  cose 


gives  cos  A 


sin  d  sin  c 


By  §36,  cosyi  =  i— 2  sin"^^ 

,,,.  ,  „,    .     cos^— cos^  cose 

Whence      i  — 2sm''t^= .    ,    . , 

sm  a  sin  c 

.  „i    .     cos/^cose+sin^  sin  e— cos<a! 

or  sm  iA  = : — ; — : , 

2  sm  a  sin  c 


103  SPHERICAL    TRIGONOMETRY 

_cos{d—c)—cosa 
~      2sin^  sin^      ' 

.    a+d—c    .    a—b+c 

sin sin 

2 2 

~  sin  b  sin  c  \^  ) 

Putting 

a-\-b-\'C        ^,       a-{-b—c                  ,  a  —  b  +  c  . 
=s,  then =s—c,  and =s—b, 

2  2  2  ' 


.    ,    .        /smis—b)  smis—c) 

we  have  s\n±A=\J •    i    •     ' 

V  sin  ^  sin^: 

Since,  also,        cos A=.i+2  cos^^A, 

we  have,  similarly, 


cos 


_,    .  _     /s'ms  s,\n{s—d) 
~"  *       sin  b  sin  c 


Hence  tanj^:^.  Aj^(^ -»)'*'»  (^-^).  m 

V        gins  sin  (s— a)  ^  ^ 

By  a  like  process,  formula  (2)  reduces  to 


/    — eo^Seo»(S—A) 


i^;?.  If,  in  formula  I,  we  advance  one  letter,  we  have 


tan*  B= .  /sin(.y— ^)sin(^-tf)^ 
V      sin^  sin  (s—b) 
And  dividing  tan-^y^  by  tan^^,  and  reducing,  we  obtain 
tan^yi      sin(j  — ^) 
tan  ^^~  sin  (^—«)' 

By  composition  and  division, 

tan^^+tan^^      sin{s—b)  +  s\n{s—a) 
tan^A—tan^B"  sin  (j— ^)— sin  (s—a)' 

By  §§  30,  38,  this  becomes 

filn^{A-\-B)  tan^c 


sin  ^A  —  B)      tan  ^  (a  —  6) ' 


(III) 


OBLIQUE-ANGLED    TRIANGLES  103 

Multiplying  \.2^\\A  by  tan^^,  and  reducing,  we  obtain 

tan -J  yi  tan  "I  ^      sin  (j— ^r) 
I  ~      sin  J     * 

By  division    and    composition,  and   by  §§  30,  38,  this   be- 
comes 

co8-|(^4--B)  _      tan^c  .^^ 

co8-|(4— B)~  tan-|(a  +  6)"  ^      ' 

Proceeding  in  a  similar  way  with  formula  II,  we  obtain 
sin  I"  (a +  6)  cot-|C 


sin  |- (a  — 6)      tan  |- (^1  —  !>) ' 


(V) 


A  d  cos-|(CT  +  &)  cot^C  .     ,-^j, 

^08^(a— 6)"~tan^(J[-j-B)'  ^      ^ 

03»  In  the  spherical  triangle  ^^C",  suppose  CD  drawn  per- 
pendicularly to  AB,  then,  by  the  formulas  for  right  spher- 
ical triangles, 


D 


In  triangle  ACD,  sin  /=sin  ^  sin  A. 

In  triangle  BCD,  sin  /  —  sin  a  sin  B. 

Whence  sin  a  sin  ^=sin  b  sin  A^ 

sin  a     sin  6  /TrxTN 

Remark.— \{  (A-|-B)>i8o°  then  {a -\- d)> i^cP,  and  if  (A-|-B)< 
180P,  then  (« -|-<J)<  180°. 


lO^. 


SPHERICAL    TRIGONOMETRY 


94,  All  cases  of  oblique-angled  triangles  may  be  solved 
by  applying  one  or  more  of  the  formulas  I,  II,  III,  IV,  V, 
VI,  VII,  as  shown  in  the  following  cases. 

CASES 

(i.)  Given  three  sides,  to  find  the  angles. 

Apply  formula  I.     Check:  apply  V  or  VI. 

(2.)  Given  three  angles,  to  find  the  sides. 

Apply  formula  II.     Check  :  apply  III  or  IV. 

(3.)  Given  two  sides  and  the  included  angle. 

Apply  V  and  VI.  and  VII.     Check  :  apply  III  or  I V. 

(4.)  Given  two  angles  and  included  side. 

Apply  III  and  I V,  and  VII.     Check :  apply  V  or  VI. 

(5.)  Given  two  angles  and  an  opposite  side. 
Apply  VII,  V,  and  III.     Check:  apply  IV. 

(6.)  Given  two  sides  and  an  opposite  angle. 
Apply  VII,  V,  and  I V.     Check  :  apply  III 


EXAMPLE — CASE   (l) 
95.  Given  a  =  81°  10'  <5  =  6o°2o'  c—\i2°2S' 

To  find  A,  B,  and  C. 


/j=  81°  10' 
b  =  60°  20' 
f  =112°  25' 

2^=253°  55' 

J  =  126°  57'  30" 

j-a=45°  47'  30" 
s—b=ttP  37'  30" 
j-f  =  i4°32'  30" 
log  sin  5=9.90259 
log  sin(5— rt)=9.85540 
log  sin(j— *)=9.9628l 
log  sin(j  —  f)=:9  39982 


To  f  ml  A. 


tan 


V       sin 


■b)  s\n{s—c) 


sin  J  sin(j— fl) 
log  sin(j— 3)=9.9628i 
log  sin(j— f)=g.39982 

colog  sin  j=o.  T4460 

colog  sin(j— <7)=o.0974i 

2)19.60464 

log  tan ^^=      9.80232 

iA=32°  23'  19" 

^=64^46'  38" 


OBLIQUE-ANGLED   TRIANGLES 


To  find  B. 

V       sin  J  sin  ( J— ^) 

log  sin  (j—rt)  =  9. 85540 

log  sin  (i-—f)  =  9. 39982 

colog  sinj-=o.0974i 

colog  sin(j'— /')  =0.03719 

2)19.38982 

log  tan  \  B^z      g. 69491 

\B=it°i\'    6" 

.5=52°  42'  12" 


To  find  C. 

tan  A  C=  .  /sin(.^-^)^i"(^-/0 , 

V        sin  J  sin(j— f) 

log  sin  (j—<i)=:9. 85540 

log  sin  (j—(J)=g. 96281 

colog  sin  J =0.0974 1 

colog  sin (j  —  f)=o. 60018 

2)20.51580 

log  tan  J  C=       10.25790 

i  C=  61°    5'  32" 

C=I22°  11'     4" 


Check. 


Formula  V,  cot  *  c = ^ — : — 7-, 7; . 

^=64°  46'  38" 
.5=52°  42'  12" 


A-B=\2° 

a=8i°  10' 

i  =60°  20' 
a  +  f>=i^i°  30' ;  i(a+3)=7o°  45' 
a—d=  20°  50';  ^{u  —  6)=io°  2s' 


4'  26" 

2'  13" 

log  tan ^M—.5)=9. 02430 

log  sin  Hrt  +  /^)=9  97501 

colog  sin^(rt  —  ^)  =  0.74279 

cot^  C=  9. 742 10 

i  C=  61°    5' 

C=I22°  11' 


EXAMPLE — CASE   (3) 
96.  Given  «  =  78°  1 5'  (5  =  56°  20' 

To  find  A,  B,  and  r. 

^(a  +  ^)=67«'  17'  30" 
|(rt-^)=io°  57'  30" 
i  C=6o° 

Formula  V  may  be  written 

tan  A  (/4  +.5)=  — ^ — ,,    ,  ., 

*  cos  ^(a  +  />) 

log  cos^(rt'  — ^)=  9.99201 

logcot^C=   9.76144 

colog  cos ^ («  +  /')=  0.41337 


log  tan ^(^+.5)=  10. 16682 

i(/f+^)  =  55°44'36"- 
^{A-B)=  6°  47'    4" 
.^=62°  31'  40" 

^=48°  57' 32"- 


C=I200 


log  sin^(rt  +  ^)=g. 96498 
log  cos^(rt  +  (^)=9.58663 
logsin|(rt-/^)=g.278g7 
log  cos^(rt— ^)=9.992oi 
logcot^C=g.76i44 

To  find  i  {A- B). 
Formula  VI  may  be  written 


^^  sin^(a  +  ^) 


log  sin^(rt  — /^)=9.27897 

log  cot  i  C=9. 76144 

colog  sin  |(rt  +^)  =0.03502 

J(^-^)=6°57'4' 


io6 


SPHERICAL    TRIGONOMETRY 


To  find  c. 
From  Fonnula  VII,  sin  f= 


sin  b  sin  C 


sin^ 


log  sin  3  =9.92027 

logsinC=9.93753 

colog  sin  .5=0.12249 

log  sin  f =9. 98029 
<r=i07°8' 


tan^( 


Check. 
Formula  III  may  be  written 

logsini(^+^)=  9-91725 

log  tan ^{a  —  b)=  9. 28696 

colog  sin  ^(.^ —i*)=  0.92762 

log  tan ^f=  10. 1 3183 

\c=  53°  33'  56"- 
^=107°    7'  51"  — 
(Discrepancy  due  to  omitted  decimals  ) 


AMBIGUOUS  CASES 

97.  (I.)  Two  sides  and  an  angle  opposite  one  of  them  are  the 
given  parts. 

If  the  side  opposite  the  given  angle  differs  from  go^  more  than  the 
other  given  side,  the  given  angle  and  the  side  opposite  beittg  either  both 
less  or  both  greater  than  90°,  there  are  two  solutions. 


(2.)  Two  angles  and  a  side  opposite  one  of  them  are  the  given  parts. 

If  the  angle  opposite  the  given  side  differs  from  90°  more  than  the 
other  given  angle,  the  given  side  and  the  angle  opposite  being  either 
both  less  or  both  greater  than  go°,  there  are  two  solutions. 

Remark. — There  is  no  solution  if,  in  either  of  the  formulas, 


sinj5= 


sin  A  %\wb 


sin  b  sin  A 


sin  a       '  sin  ^ 

the  numerator  of  the  fraction  is  greater  than  the  denominator. 


OBLIQUE-ANGLED    TRIANGLES 


lorj 


EXAMPLE— CASE   (6) 
98,  Given  ^=40°  16'  <5=47°44'  ^=52°  30' 

To  find  B,  B',  C,  C.  and  c,  c'. 


To  find  B  and  B'. 

Formula  VII  may  be  written 

sin^  sin<5 

sin  B=^ : 

sin  a 

log  sin  A=g.  89947 

log  sin  ^=9. 86924 

colog  sin  rt=o.  18953 

log  sin  iff =9.95824 

B=i  65°  16'  30" 

^'  =  114°  43'  30" 

To  find  c. 
Formula  IV  may  be  written 

^^    .    _cosH/f+ig)tanH^  +  ^) 
*^"  *  '-  cosM^-^) 

log  cos  ^(^  +  y^)=9. 71326 
log  tan^(«  +  (^)=9.98484 

colog  C0S^(/4— ^)=O.CX)270 

l^g  tan  ^(r= 9. 70080 

^c=26°  39'  42" 
^=53°  19'  24" 

To  find  c\ 
log  cos^(^+i5')=9.0463l 
log  tan^(a  +  <^)=9.98484 
colog  cos^(yi  — ^')=o.o6745 

log  tan^f'=9.0986o 

\c'=  7°    g'    9" 
<:'  =  14°  18'  18" 


To  find  C. 
Formula  V  may  be  written 

,^     sini(rt  +  ^)  tanA(^— 5) 
^  sin^(a-<!>) 

\og%\n^{a  +  b)=  9.84177 
log  tan^(^— ^)=  9.04901  n 
colog  sin ^(rt  — (5)=  1. 18633  n 
log  cot  ^C=  10.077 1 1 
iC=39°56'24" 
C=790  52'  48" 

To  find  C. 

logsin^(rt  +  ^)=  9,84177 

log  tan  ^(^-^')=  9-781530 

colog  sin  ^  (a  — 3)=   1. 18633  n 

log  cot  ^  C"  =  10. 80963 

lc'=  8°  48' 41" 
C'=I70  37'22" 

Check. 
Formula  III  may  be  written 

.     ,     sin  ^  sin  f 

sm  b= r-7; 

sine 

log  sin .5=9.95824 

log  sinr=9. 90418 

colc^  sin  C"=o.oo682 

log  sin  <J= 9. 86924 

^=47°  44' 


EXERCISES 

99,  (I.)  In  the  spherical  triangle  ABC,  the  side  a=  124°  53',  the 
side  b  =  31°  19',  and  the  angle  A  =  16°  26'.     Find  the  other  parts. 

(2.)  In  the  oblique-angled  spherical  triangle  ABC,  angle  A  =  128° 
45',  angle  C=  30°  35',  and  the  angle  .ff  =  68°  50',     Find  the  other  parts. 


*  The  letter  "  n  "  indicates  that  these  quantities  are  negative. 


io8  SPHERICAL    TRIGONOMETRY 

(3.)  In  the  spherical  triangle  ABC,  the  side  ^  =  78°  15',  ^=56°  20', 
and  A  =  120°.     Required  the  other  parts. 

(4.)  In  the  spherical  triangle  ABC,  the  angle  ^4  =  125°  20',  the  an- 
gle C=48°  30',  and  the  side  ^  =  83°  13'.  Required  the  remaining 
parts. 

(5.)  In  the  spherical  triangle  ABC,  the  side  f  =  40°  35',  (5  =  39*^  10', 
and  «  =  71°  15'.     Required  the  angles. 

(6.)  In  the  spherical  triangle -r4^C  the  angle  ^  =  109°  55',  ^=116'' 
38',  and  C=  120°  43'.     Required  the  sides. 

(7.)  In  the  spherical  triangle  ABC,  the  angle  ^  =  130°  5'  22",  the 
angle  C=36°45'  28",  and  the  side  (J  =  44°  13'  45".  Required  the  re- 
maining parts. 

(8.)  In  the  spherical  triangle  ABC,  the  angle  ^  =  33°  15'  7",  B-= 
31°  34'  38",  and  C=  161°  25'  17".     Required  the  sides. 

(9.)  In  the  spherical  triangle  ABC,  the  side  c=.\\2°  22'  58",  b-=. 
52°  39'  4",  and  a  =  %(f  16'  53",     Required  the  angles. 

(10.)  In  the  spherical  triangle  ABC,  the  side  €^=.76°  35'  36",  h  = 
50°  10'  30",  and  the  angle  ^^  =  34°  15'  3".  Required  the  remaining 
parts. 

AREA  OF  THE  SPHERICAL  TRIANGLE 

100,  It  is  proved  in  geometry  that  the  area  of  a  spherical 
triangle  is  equal  to  its  spherical  excess,  that  is, 
area  =  (.<4  +  ^+  C—  2  rt.  angles)  X  area  of  the  tri-rectangular  triangle, 
where  A,  B,  and  C  are  the  angles  of  the  spherical  triangle. 
Hence 

area v4 -f- i? -j- (7— 1 80° 

surface  of  sphere  ~  720° 

The  surface  of  the  sphere  is  ^nrR*,  therefore 

<^-|-B  +  C— I8OA 


area 


^irB'y- 


180°  / 

The  following  formula,  called  Lhuilier's  theorem,  simpli- 
fies the  derivation  of  {A-^B-\-C—i2>0°)  where  the  three 


OBLIQUE-ANGLED    TRIANGLES  109 

sides  of  the  spherical  triangle  are  given ;  in  it  a,  b,  and  c 
denote  the  sides  of  the  triangle,  and  2s  =  a-\-b  +  c. 
tan  /^+g+g-180°\  =  yt«n  i  s  tan  i  (_s-a)  tan  i(s-6)  tan  i  (s-c). 

EXERCISES 

(I.)  The  angles  of  a  spherical  triangle  are,  ^^  =  63°  5=84"  21', 
C=79°;  the  radius  of  the  sphere  is  10  in.  What  is  the  area  of  the 
triangle  ? 

(2.)  The  sides  of  a  spherical  triangle  are,  a  =  6.47  in.,  /5  =  8.39  in., 
f  =  9.43  in. ;  the  radius  of  the  sphere  is  25  in.  What  is  the  area  of 
the  triangle  } 

(3.)  In  a  spherical  triangle,  ^  =  75°  16',  ^  =  39°  20',  c  =  26  in.;  the 
radius  of  the  sphere  is  14  in.     Find  the  area  of  the  triangle. 

(4.)  In  a  spherical  triangle,  a  =  441  miles,  ^  =  287  miles,  C-=z^2P  21'; 
the  radius  of  the  sphere  is  3960  miles.    Find  the  area  of  the  triangle. 


CHAPTER  X 

APPLICATIONS  TO  THE  CELESTIAL  AND  TERRES- 
TRIAL SPHERES 

ASTRONOMICAL  PROBLEMS 

101.  An  observer  at  any  place  on  the  earth's  surface 
finds  himself  seemingly  at  the  centre  of  a  sphere,  one-half 
of  which  is  the  sky  above  him.  This  sphere  is  called  the 
celestial  sphere,  and  upon  its  surface  appear  all  the  heavenly 
bodies.  The  entire  sphere  seems  to  turn  completely  around 
once  in  23  hours  and  56  minutes,  as  on  an  axis.  The  im- 
aginary axis  is  the  axis  of  the  earth  indefinitely  produced. 
The  points  in  which  it  pierces  the  celestial  .sphere  appear 
stationary,  and  are  called  the  north  and  south  poles  of  the 
heavens.  The  North  Star  (Polaris)  marks  very  nearly  (with- 
in 1°  16')  the  position  of  the  north  pole.  As  the  observer 
travels  towards  the  north  he  finds  that  the  north  pole  of  the 
heavens  appears  higher  and  higher  up  in  the  sky,  and  that 
its  height  above  the  horizon,  measured  in  degrees,  corre- 
sponds to  the  latitude  of  the  place  of  observation. 

The  fixed  stars  and  nebulae  preserve  the  same  relative 
positions  to  each  other.  The  sun,  moon,  planets,  and  com- 
ets change  their  positions-  with  respect  to  the  fixed  stars 
continually,  the  sun  appearing  to  move  eastward  among 
the  stars  about  a  degree  a  day,  and  the  moon  about  thir- 
teen times  as  far. 


AP PLICA  TIONS  1 1 1 

The  zenith  is  the  point  on  the  celestial  sphere  directly 
overhead. 

The  horizon  is  the  great  circle  everywhere  90°  from  the 
zenith. 

The  celestial  equator  is  the  great  circle  in  which  the 
plane  of  the  earth's  equator  if  extended  would  cut  the  ce- 
lestial sphere. 

The  ecliptic  is  the  path  on  the  celestial  sphere  described 
by  the  sun  in  its  apparent  eastward  motion  among  the  stars. 
The  ecliptic  is  a  great  circle  inclined  to  the  plane  of  the 
equator  at  an  angle  of  approximately  23^°. 

The  poles  of  the  equator  are  the  points  where  the  axis 
of  the  earth  if  produced  would  pierce  the  celestial  sphere, 
and  are  each  90°  from  the  equator. 

The  poles  of  the  ecliptic  are  each  90°  from  the  ecliptic. 

The  equinoxes  are  the  points  where  the  celestial  equa- 
tor and  ecliptic  intersect ;  that  which  the  sun  crosses  when 
coming  north  being  called  the  vernal  equinox,  and  that 
which  it  crosses  when  going  south  the  autumnal  equinox. 

The  declination  of  a  heavenly  body  is  its  distance,  meas- 
ured in  degrees,  north  or  south  of  the  celestial  equator. 

The  right  ascension  of  a  heavenly  body  is  the  distance, 
measured  in  degrees  eastward  on  the  celestial  equator,  from 
the  vernal  equinox  to  the  great  circle  passing  through  the 
poles  of  the  equator  and  this  body. 

The  celestial  latitude  of  a  heavenly  body  is  the  dis- 
tance from  the  ecliptic  measured  in  degrees  on  the  great 
circle  passing  through  the  pole  of  the  ecliptic  and  the 
body. 

The  celestial  longitude  of  a  heavenly  body  is  the  dis- 
tance, measured  in  degrees  eastward  on  the  ecliptic,  from 


112  SPHERICAL    TRIGONOMETRY 

the  vernal  equinox  to  the  great  circle  passing  through  the 
pole  of  the  ecliptic  and  the  body. 

EXERCISES 
(i.)  The'Vi^ht  ascension  "of  a  given  star  is  25°  35',  and  its  declina- 
tion is  -f  (north)  63°  26'.     Assuming  the  angle  between  the  celestial 
equator  and  the  ecliptic  to  be  23°  27',  find  the  celestial  latitude  and 
celestial  longitude. 


In  this  figure  AB  is  the  celestial  equator,  AC  the  ecliptic,  P  the  pole  of 
the  equator,  /"  the  pole  of  the  ecliptic.  S  is  the  position  of  the  star,  and 
tlie  lines  SB  and  SC  are  drawn  through  P  and  P'  perpendicular  to  ^^and 
AC.  AB  is  the  right  ascension  and  BS  the  declination  of  the  star,  while 
AC\s  the  longitude  and  SC  the  latitude  of  the  star. 

In  the  spherical  triangle  P' PS,  it  will  be  seen  that  PS  is  the  comple- 
ment of  the  celestial  latitude,  PS  the  complement  of  the  declination,  and 
P PS  is  90°  plus  the  right  ascension.  It  is  to  be  noted  that  A  is  the  ver- 
nal equinox. 

(2.)  The  declination  of  the  sun  on  December  21st  is  — (south) 
23°  27'.  At  what  time  will  the  sun  rise  as  seen  from  a  place  whose 
latitude  is  41°  18'  north  } 

The  arc  ZS  which  is  the  distance  from  the  zenith  to  the  centre  of  the  sun 
when  the  sun's  upper  rim  is  on  the  horizon  is  90°  50'.  The  50'  is  made  up 
of  the  sun's  semi-diameter  of  16',  plus  the  correction  for  refraction  of  34'. 


AP  PLICA  TIONS 


"3 


(3.)  The  declination  of  the  sun  on  December  21st  is  —  (south) 
23°  27'.  At  what  time  would  the  sun  set  as  seen  from  a  place  in  lati- 
tude 50°  35'  north  ? 


In  these  figures  P  is  the  pole  of  the  equator,  Z  the  zenith,  EQ  the  celes- 
tial equator.  AS  is  the  declination  of  the  sun,  ZS=cfP  5q',  /'5=90°+dec- 
lination,  /'Z=go°— latitude.  The  problem  is  to  find  the  angle  SPZ.  An 
angle  of  15°  at  the  pole  corresponds  to  i  hour  of  time. 

GEOGRAPHICAL   PROBLEMS 

102,  The  meridian  of  a  place  is  the  great  circle  passing 
through  the  place  and  the  poles  of  the  earth. 

The  latitude  of  a  place  is  the  arc  of  the  meridian  of  the 
place  extending  from  the  equator  to  the  place. 

Latitude  is  measured  north  and  south  of  the  equator  from  0°  to  go°. 

The  longitude  of  a  place  is  the  arc  of  the  equator  extend- 
ing from  the  zero  meridian  to  the  meridian  of  the  place. 
The  meridian  of  the  Greenwich  Observatory  is  usually  taken 
as  the  zero  meridian. 

Longitude  is  measured  east  or  west  from  0°  to  180°. 
The  longitude  of  a  place  is  also  the  angle  between  the  zero  meridian  and 
the  meridian  of  the  place. 


114  SPHERICAL    TRIGONOMETRY 

In  the  following  problems  one  minute  is  taken  equal  to  one  geo- 
graphical mile. 

(I.)  Required  the  distance  in  geographical  miles  between  two 
places,  D  and  E,  on  the  earth's  surface.  The  longitude  of  D  is  60° 
15'  E.,  and  the  latitude  20°  10'  N.  The  longitude  of  E  is  115°  20'  E., 
and  the  latitude  37°  20  N. 


In  this  figure  A  C  represents  the  equator  of  the  earth,  P  the  north  pole, 
and^  the  intersection  of  the  meridian  of  Greenwich  with  the  equator.  PB 
and  PC  represent  meridians  drawn  through  D  and  E  respectively.  Then 
AB'\%  the  longitude  and  BD  the  latitude  of  Z) ;  AC  the  longitude  and  CE 
the  latitude  of  E. 

(2.)  Required  the  distance  from  New  York,  latitude  40°  43'  N., 
longitude  74°  o'  W.,  to  San  Francisco,  latitude  37°  48'  N.,  longitude 
1 22°  28'  W.,  on  the  shortest  route. 

(3.)  Required  the  distance  from  Sandy  Hook,  latitude  40°  28'  N., 
longitude  74°  i'  W.,  to  Madeira,  in  latitude  32°  28'  N.,  longitude  16°  55, 
\V.,  on  the  shortest  route. 

(4.)  Required  the  distance  from  San  Francisco,  latitude  37*^  48' 
N.,  longitude  122°  28'  W.,  to  Batavia  in  Java,  latitude  6°  9'  S.,  longi- 
tude 106°  53'  E.,  on  the  shortest  route. 

(5.)  Required  the  distance  from  San  Francisco,  latitude  37°  48' 
N.,  longitude  122°  28'  W.,  to  Valparaiso,  latitude  33°  2'  S.,  longitude 
71°  41'  W.,  on  the  shortest  route. 


CHAPTER   XI 

GRAPHICAL   SOLUTION   OF   A   SPHERICAL  TRIANGLE 

103.  The  given  parts  of  a  spherical  triangle  may  be  laid 
off,  and  then  the  required  parts  may  be  measured,  by  making 
use  of  a  globe  fitted  to  a  hemispherical  cup. 

The  sides  of  the  spherical  triangle  are  arcs  of  great  circles, 
and  may  be  drawn  on  the  globe  with  a  pencil,  using  the 
rim  of  the  cup,  which  is  a  great  circle,  as  a  ruler.  The  rim 
of  the  cup  is  graduated  from  o°  to  i8o°  in  both  directions. 

The  angle  of  a  spherical  triangle  may  be  measured  on  a 
great  circle  drawn  on  the  sphere  at  a  distance  of  90°  from 
the  vertex  of  the  angle.* 

Case  I.  Given  the  sides  a,  b,  and  c  of  a  spherical  triangle^ 
to  determine  the  angles  A ,  B,  and  C. 

Place  the  globe  in  the  cup,  and  draw  upon  it  a  line  equal 
to  the  number  of  degrees  in  the  side  c,  using  the  rim  of  the 
cup  as  a  ruler.  Mark  the  extremities  of  this  line  A  and  B. 
With  A  and  B  as  centres,  and  b  and  a  respectively  as  radii, 
draw  with  the  dividers  two  ares  intersecting  at  C  (Fig.  i). 
Then,  placing  the  globe  in  the  cup  so  that  the  points.^  and 
C  shall  rest  on  the  rim,  draw  the  line  AC=b,  and  in  the 
same  way  draw  BC—a. 

To  measure  the  angle  A  place  the  arc  AB  in  coincidence 

*  Slated  globes,  three  inches  in  diameter,  made  of  papier-mache,  and  held 
in  metal  hemispherical  cups,  are  manufactured  for  the  use  of  students  of 
spherical  trigonometry  at  a  small  cost. 


n6 


SPHERICAL  TRIGONOMETRY 


with  the  rim  of  the  cup,  and  make  AE  equal  to  90"^.  Also 
make  AF  '\n  AC  produced  equal  to  90°.  Then  place  the 
globe  in  the  cup  so  that  E  and  /^  shall  be  in  the  rim,  and 
note  the  measure  of  the  arc  EF.  This  is  the  measure  of  the 
angle  A.  In  the  same  way  the  angles  B  and  C  can  be  de- 
termined. 


Case  II.  Given  the  angles  Ay  B,  and  C,  to  find  the  sides 
a,  by  and  c. 

Subtract  A,  B,  and  C  each  from  180°,  to  obtain  the  sides 
a\  b',  and  c'  of  the  polar  triangle.  Construct  this  polar  tri- 
angle according  to  the  method  employed  in  Case  I.  Mark 
its  vertices  A',  B',  and  C.  With  each  of  these  vertices  as 
a  centre,  and  a  radius  equal  to  90°,  describe  arcs  with  the  di- 
viders. The  points  of  intersection  of  these  arcs  will  be  the 
vertices  A,  B,  and  C  of  the  given  triangle.  The  sides  of 
this  triangle  a,  <^,  and  c  can  then  be  measured  on  the  rim 
of  the  cup. 


GRAPHICAL  SOLUTION 


"7 


Case  III.  Given  two  sides,  b  and  c,  attd  the  included  angle 
A,  to  find  B,  C,  and  a. 

Lay  off  (Fig.  3)  the  line  AB  equal  to  c,  and  mark  the 
point  D  in  AB  produced,  so  that  AD  equals  90°.  With  the 
dividers  mark  another  point,  F,  at  a  distance  of  90°  from  A. 
Turn  the  globe  in  the  cup  till  D  and  ^are  both  in  the  rim, 
and  make  DE  equal  to  the  number  of  degrees  in  the  angle  A. 
With  A  and  E  in  the  rim  of  the  cup,  draw  the  Ym^  AC  equal 
to  the  number  of  degrees  in  the  side  b.  Join  C  and  B.  The 
required  parts  of  the  triangle  can  then  be  measured. 


Case  IV.  Given  the  angles  A  and  B  and  the  included  side 
c,  to  find  a,  b,  and  C. 

Lay  off  the  line  AB  equal  to  c.  Then  construct  the  given 
angles  at  A  and  B,  as  in  Case  IIL,  and  extend  their  sides  to 
intersect  at  C. 

Case  V.  Given  the  sides  b,  a,  and  the  angle  A  opposite  one 
of  these  sides,  to  find  c,  B,  and  C.     (Ambiguous  case.) 


Il8  SPHERICAL    TRIGONOMETRY 

Lay  off  (Fig.  4)  yi  6"  equal  to  b,  and  construct  the  angle  A 
as  in  Case  III.  Take  c  in  the  dividers  as  a  radius,  and  with 
6^  as  a  centre  describe  arcs  cutting  the  other  side  of  the  tri- 
angle in  B  and  B\  and  measure  the  remaining  parts  of  the 
two  triangles. 

If  the  arc  described  with  C  as  a  centre  does  not  cut  the  other  side  of  the 
triangle,  there  is  no  solution'.     If  tangent,  there  is  one  solution. 

Case  VI.  Given  the  angles  A,  B,  and  the  side  a  opposite 
one  of  the  angles. 

Construct  the  polar  triangle  of  the  given  triangle  by 
Case  v.;  then  construct  the  original  triangle  as  in  Case  II., 
and  measure  the  parts  required. 

The  constructions  given  above  include  all  cases  of  right  and  quadrantal 
triangles. 


CHAPTER   XII 
RECAPITULATION    OF    FORMULAS 

ELEMENTARY    RELATIONS   (§   lO) 

sinjr  cosx 

tan  X  = ,  cot  X  =  ■ 


secjr: 


cos^  sin;ir 

I  1 


cos;r  sin-r 

tan  X  cot  x=^i, 
sin"  ^  +  cos'' ;r  =  i,' 
I  +  tan''  X  =  sec^  x, 
I  +  cot''  X  =.  esc"  X. 

RIGHT    TRIANGLES    (§§  I4  AND   27) 

a  .  o 

sinA  =  -,  sinB  =  -, 

c  c 

b  a 

cos  A  =  -,  cos  B  =  - , 

c  c 

a  ^      b 

tan  y€  =  T ,  tan  B  =  ~, 

b  a 

.    -<5  „     ^ 

cot  ^  =  - ,  cot  B  =  1, 

a  o 

a'^b'^  —  c', 
where  r  =  hypotenuse,  a  and  b  sides  about  the  right  angle;  A  and  ^ 
the  acute  angles  opposite  a  and  b. 

FUNCTIONS  OF  TWO  ANGLES  (§§  3O-34) 

sin  (Jr-}-/)  =  sin  jr  cosj'  +  cos;^  sin^, 
sin  (.r — y)=.?\nx  cosj  — cos  x  €\ny, 
cos  (x -f-j)  =  cos  X  cos/  —  sin  ;r  sin_y, 
cos  {x — /)  =  cosjr  cos/-|-sin:r  sin_y. 


I20  RECAPITULATION  OF  FORMULAS 

tan;r+tan>' 


tan(jr+_y)  = 
tan  {x—y)  = 
cot(jr+/)  = 
cot  (x—y)  = 


I  — tanx  tan^* 
tan  jr  —  XsMy 

i+tan^  tanj' 

cot  JT  cot^ —  I 
cotj+cot;r  ' 

cotx  COt>'+  I 
cot_y  —  COt;ir 


FUNCTIONS  OF  TWICE  AN  ANGLE  (§  36) 


Sin  2-r  =  2  sin  JT  cos;r, 
COS  2x  =  cos'  X — sin'  x, 
=  1  — 2  sin'jr, 
=  2  cos"  jr — I, 
2  tan  jr 


tan  2x=z- 


I  — tan^'x 

cot'x —  I 
cot  2X  = .      , 

2  cot  jr 
FUNCTIONS   OF  HALF  AN  ANGLE  (§  37) 


sin  i;f  =  d:\/- 


,          .      /i+cosx 
cos  ^x  =  ±v/  — , 


tan  ix  =  ±:K/ — 
■*       V  I— « 


+  cosjr 
cot 


SUMS  AND   DIFFERENCES  OF  FUNCTIONS  (§  38) 

sin  «-}-sin7/  =  2  sin^{u-\-v)  cos^(u  —  v), 

sin«  — sin?/  =  2  cos^{u-{-v)  sin  ^{u — v), 
cos  u  -\-  cos  7/  =  2  cos  i  («  +  7'')  cos  4  («  —  'Z'), 
cosu  —  cos 7/  =  — 2  sin  4(«  +  «')  sin  ^{u — v). 

s\nu-{-s\nv  __  tan  ^(u-{-v) 

sin  «  —  sin  v~  tan  ^{u  —  v)' 


RECAPITULATION  OF  FORMULAS  I2f 

OBLIQUE   TRIANGLES   (§§  42-45) 

a _s\x\  A  a      sin>4  b      sin  ^ 

b~%\nB''  r~sinC'  ^"~sinC" 

a  —  b  _\.2iXs.\{A  —  B) 
a-\-b~'X.2.n\{^A-\-By 
a  —  c     tan  \{A  —  C) 
a"+7  ~  tanT(Z+Tj' 
b  —  c      X.2L.n\{B  —  C) 
b-\-c~X.2.n\{B-^Cy 

a^  =  b'^  -\- c"^  —  2bc  cos  A, 
b"^  z=.  c^  -^  a^  —  2ca  cos  B, 
c''  =  a''-\-b''  —  2abcosC. 

tani^=\/EMEl. 
s(s  —  a) 

s{s  —  b) 

s(s — c) 

a  +  b-\-c 
where  s=^ 

2 

tani^^jH^-  taniff=j3^.  taniC  =  j3^. 

where  A-=V^^^^-^)^^-^^ 

AREA   OF  A   TRIANGLE  (§  46) 

S-=-\ac  sin  B.      S^\ba  sin  C.      S=-\cb  sin  ^. 
5= -y/^  (j  —  rt)  (j -^  ^)  (j  —  c). 

LOGARITHMIC,  COSINE,   SINE,  AND  EXPONENTIAL  SERIES 

(§  58) 

log^(i+-*")=-^— -7  +  --  — +.  etc. 

X^        X*        X* 

cos^=i-^+--g^+,etc. 


122  RECAPITULATION  OF  FORMULAS 

sin^  =  ^-^j4-j--^+.etc. 

X^        X^        X* 

^=i+^  +  -,+  -  +  -+,etc. 

DE   MOIVRE'S  theorem  (§  71) 

(cos;r4-  v/— J  sin  jr)''=cos«x4-'\/— '  s'xnnx. 

-_,        ."  «(« — i)(«  —  2) 

sm nx  =  n  cos"     '4:  sin.r cos"~^  jr  sin^x4-,  etc. 

3! 

cosnx  =  n  cos"^ j —  cos"~*^  sin'':r+,  etc. 

2 1 

HYPERBOLIC   FUNCTIONS   (§  75) 
if*"  —  ^~-* 


sinh  jr  = 
cosh  -r  = 


2 


2 
^^  =  cos  x-\-t  sin  jf. 


ptX ^  —  tx 

sin  j:  = 


eosjr 


2Z 


sin  ix  =  — '-  =  t  sinh  x, 

2 

cos«jr  =  — - — —  =cosh;r. 


SPHERICAL  TRIANGLES 
RIGHT   AND   QUADRANTAL  TRIANGLES  (§§  83,  87) 

Use  Napier's  rules. 

OBLIQUE  TRIANGLES   (§§  89-93) 

cos^?  =  cos(i  cos<r-}-sin<5  sinr  cos^. 
cos  Az=.  —  cos  B  cos  C-j-  sin  B  sin  C  cos  a. 


tani^  =  X/^I^?^^^ "'"("- 
V      sin  J  sin  (j  —  a) 


RECAPITULATION  OF  FORMULAS  123 


^  =  \ll 


tani  — '  '        008  5  003(5-^) 


cos  (S—B)  cos(5— C) 
sin  ^iA  +  £)_      tan^c 
sin  ^{A  —  B)~~ tan  ^(a  —  b)' 
cos^(A-{-B)_      tan^f 
cosJi{A  —  B)~tRniia-[-d)' 
sin  ^  (a  +  d)  cot  J  C 


sini(«  — <^)     tani{A  —  B) 
cos  ^  («j-f_^  _       cot  i  C 
cos^{a  —  d}~X.an^{A-[-B)' 

sin  a sin  b 

sin^~sini/' 

AREA   OF   SPHERICAL  TRIANGLES   (§  lOl) 

/A  +  B+C- i8oo\ 
area  =  nR-[ ^g^, ) 

tan  (  +   —'  °  \  _  y^n  i  j  tan  i (j— a)  tan  i  {s—d)  tan  ^  (j— <r). 


APPENDIX 

RELATIONS   OF  THE   PLANE,  SPHERICAL,  AND   PSEUDO- 
SPHERICAL   TRIGONOMETRIES 

We  have  up  to  the  present  considered  the  trigonometries 
which  deal  with  figures  on  a  plane  or  spherical  surface.  A 
characteristic  feature  of  these  two  surfaces  is  that  the  curv- 
ature of  the  plane  is  zero,  while  that  of  the  sphere  is  a  posi- 
tive constant  p.  If  the  radius  of  the  sphere  is  increased  in- 
definitely, its  surface  approaches  the  plane  as  a  limit  while 
its  curvature  p  approaches  o. 

In  works  on  absolute  geometry  it  is  shown  that  there  ex- 
ists a  surface  which  has  a  constant  negative  curvature:  it  is 
called  a  pseudo-sphere,  and  the  trigonometry  upon  it  pseudo- 
spherical  trigonometry. 

We  observe  that  as  p  passes  continuously  from  positive 
to  negative  values,  we  pass  from  the  sphere  through  the 
plane  to  the  pseudo-sphere.  Thus  the  formulas  of  plane 
trigonometry  are  the  limiting  cases  of  those  of  either  of  the 
two  other  trigonometries. 

In  the  treatment  of  spherical  trigonometry  the  radius  of 
the  sphere  has  been  taken  as  unity.  If,  however,  the  radius 
of  the  sphere  is  r,  and  a,  b,  and  c  denote  the  lengths  of  the 
sides  of  the  spherical  triangle,  the  formulas  are  changed,  in 

n  h  r 

that  a  is  replaced  by  -,   b  by  -,   and  ^  by  -  ;  thus, 


126 


APPENDIX 


sin  C=- 


becomes 


sinC= 


.    c 

sm- 

r 

.    a 

sin- 

r 


The  formulas  for  pseudo-spherical  trigonometry  are  the 
same  as  the  formulas  of  spherical  trigonometry,  except  that 

the  hyperbolic  functions  of  -,    -,  and  -  are  substituted  for 

the  trigonometric. 

Thus,  corresponding  to  the  above  formula  of  spherical 
trigonometry,  is  the  formula 


sinC= 


sinh- 
r 

sinh- 
r 


of  pseudo-spherical  trigonometry. 


PSEUDO-SPHKRH 


The  pseudo-sphere  is  generated  by  revolving  the  curve  whose  equation  is 


y=r  log 


-Vr»-; 


about  its  y  axis.     The  radius  of  the  base  of  the  pseudo-sphere  is  r. 


APPENDIX  127 

Hence  the  formulas  of  plane  trigonometry  can  be  derived 
from  the  formulas  of  either  spherical  or  pseudo- spherical 
trigonometry  by  expressing  the  functions  in  series  and  al- 
lowing r  to  increase  without  limit. 

Example. — Show  that  if  r  is  increased  indefinitely  the  following 

corresponding  formulas  for  the  spherical  and  pseudo-spherical  right 

triangle 

a  b        c 

cos    =cos-cos-»  (i) 

r  r        r  ^  ' 

cosh  -  =  cosh  -  cosh  - ,  (2) 

r  r  r  ' 

reduce  to  the  corresponding  formula  for  a  plane  right  triangle;  that 

is,  to 

d'=^b''-\-c\  (3) 

Substituting  the  series  cos  - ,  etc. ,  in  equation  (i),  we  obtain 

('-r0--H-^Q"--)(-M^y--> 

2  !  >'*      4 !  r*  2  !  r*      2  !  r^      4  !  ;•*  ^ 

Substituting  in  equation  (2)  the  series  for  cosh  -  ,  etc. ,  which  we  obtain  from 

cosh  X  = ,  we  have 

2 


2  !  ;-'i      4 1  ^  2  !  r*      2  !  r      4  !  ;^ 

Cancelling  i  in  equationo  (4)  and  (5),  multiplying  by  r^,  and,  finally,  allowing 
r  to  increase  without  limit,  we  get  from  either  equation 


EXERCISES 
Derive  each  of  the  following  formulas  of  plane  trigonometry  from 
the  corresponding  formula  of  spherical  trigonometry,  and  also  fron: 
the  corresponding  formula  of  pseudo-spherical  trigonometry. 


128  APPENDIX 

Right  triangles  ;  A  z^  right  angle* 

(I.)  Plane,  sin  C=  — 

a 

Spherical,  sin  C=—. — • 

^  sin  a 

Pseudo-spherical,        sin  C=.-r-r — 

Oblique  Triangles. 
(2.)  Plane,  a'  —b'-\'c'  —  2  be  cos  A. 

Spherical,  cos  a  =  cos  b  cos  ^+  sin  3  sin  c  cos  A. 

Pseudo-spherical,  cosh  a  =  cosh  ^  cosh£-  +  sinh^  sinhtr  cos^. 


(3.)  Plane,  5  =  -/^  (j— a)  (j— <J)  i^s—c). 

Spherical, 

tan ^    ^     ^ ^  =  >/tan  ^  -  tan  i—;:-  tan  i -^t"  tan  i-^— 

Pseudo-spherical, 

tan  5^ ^=Vtanh  4  -  tanh  4^ •  tanh  4  ^ tanh  \ -. 

4  ^  *  r  *     r  '     r  "r 


ANSWERS   TO    EXERCISES 


§  4  (page  3). 

(I.)  192°  Si'25f". 
Quadrant  III. 
(2.)  250. 
(3.)  2870,  647=. 
(4.)  Quadrant  III. 

§  9  (page  g). 

tan  icxx)°  is  negative, 
cos  810°  is  o. 
sin  760°  is  positive, 
cot  —  70°  is  negative, 
cos  —  550^  is  negative, 
tan  —  560°  is  negative, 
sec  300°  is  positive, 
cot  1 560°  is  negative, 
sin  130°  is  positive, 
cos  260°  is  negative, 
tan  310°  is  negative. 

§  13  (page  11). 

(3.)  cos  —  30°  =  ^  V3- 
tan  -30°  =  -^  v/3. 
cot  —  30°  =  —  Y^3, 
sec  —  30°  =  J  ^/^y, 
CSC  —  30°  =  —  2. 

(4.)  cos;r=  —  I  ■\/2, 

tan  X  =  \  ■\/2, 

cot  .r  =  2  \/2, 

sec  .r  =  —  I  sjz, 
cscr  =  —  3. 
9 


(5.)  cos^  =  |,    tan/  =  — f, 

cot  J  =  —  J,    sec/  =  |, 

csc/  =  — ^. 
(6.)  sin  60°  =  ^  ■\/3, 

tan  60°=  \/3, 

cot  60°  =  i  y/ 1, 

sec  60°  =  2, 

CSC  60° = f  y  3. 

(7.)  cos  0°  =  I,    tan  0°  =  o. 
(8.)  sin2-  =  f,    cos  .3'  =  I, 
cot  2-  =  |,    sec  z  =  %, 

CSC  5"=  |. 

(9.)  sin  45°  =  cos45°  =  i  ^2, 
tan  45°=  I, 

sec  45°  =  CSC  45°  —  -1^2. 
(10.)  sinj  =  — ^'v/5,  cosj/  =  — f, 
cot  J/  =  f  v/5,    sec/  =  — I, 
CSC/  =  —  I  y' J. 
(II.)  sin  30°  =  i    cos  30°  =  1  ^/'^„ 

tan3o°  =  i-v/3. 
sec3o°  =  | -v/3. 

CSC  30°  =  2. 

(12.)  sinjr  =  f,    cos.r  =  — -|. 
(13.)  J\MV~S- 

§  17  (page  14). 
(I.)  sin  70°  =  cos  20°, 
cos  60°  =  sin  30°, 
cos  89°  31'=  sin  29', 
cot  47°=  tan  43°, 


130 


ANSWERS    TO  EXERCISES 


tan  63°=  cot  27°, 

sin  72°  39'=  cos  17°  21'. 

(2.)  .r  =  30°. 

<3.)  X  =  22°  30'. 

(4.)  .r=i8°. 

(5.)  .r=i5°. 

§  25  (page  21). 

(I.)  225°  and  315°, 

60°  and  240°. 
(2.)  60°   120°,  420°  480°. 
(3.)  sin  —  3o==-i, 

cos  —30°=^  -^3, 

sin  765°=  cos  765  =  ^  -y/2, 


VI 


TS' 


sin  120°= 
cos  120°: 
sin  210°=  —  ^, 

cos  210°=  —  ^  "v/s- 
(4.)   Tiie   functions   of  405°  are 

equal  to  the  functions  of  45°. 

sin  600°=  —^  x/^, 

cos  600°=  — i, 

tan  600°=  ^^3, 

cot  600°=  ^  -v/3, 

sec  600°=  —2, 

CSC  600°=  — I  -v/3. 
The  functions  of  1125°  are 

equal  to  the  functions  of  45°. 

sin  —  45°  =  — ^  \/2, 

cos  — 45°=»  V2. 

tan  —  45°=  cot  —  45°=  —  I , 

sec  — 45°=-v/2, 

CSC  — 45°=  — -v/2. 

sin  225°=  cos  225°=— ^  -v/2, 

tan  225°=  cot  225°=  I, 

sec  225°=  esc  225°=  —  v/2. 
(5.)  The  functions  of  —  120°  are 


the  same  as  those   of  600° 

given  in  (4). 

sin  —  225°  =  ^  -v/2, 

cos— 225°  =  —  kV^' 

tan  —  225°=  cot  —  225°=  —  I , 

sec  —  225°=  —  \/2, 

CSC  — 225°= -v/2, 

sin  —  420°  =  —  ^  v^S, 

cos  —  420°  =  |. 

tan  —  420°  =  —  "v/J, 

cot-420°  =  — Ji"v/3, 

sec  — 420°  =  2, 

CSC  —  420"^  =  —  t  Vs- 

The  functions  of  3270°  are 

equal  to  the  functions  of  30°. 
(6.)  sin  233°  =  —  COS  37° 

cos  233°  =  — sin  37° 

tan  233°  =  cot  37°, 

cot  233°  =  tan  27°, 

sec  233°  =  —  esc  37°, 

CSC  233°  =  —  sec  37°. 

sin  —  197-^  =  sin  17°, 

cos  —  1 970  =  —  cos  1 7° 

tan  — 197°  =  — tan  17°, 

cot—  197°  =  — cot  17°. 

sec  —  1 97"  =  —  sec  1 7° 

esc  —  1 97°  =  esc  1 70, 

sin  894='  =  sin  6° 

cos  894°  =  —  cos  6°, 

tan  894°  =  —  tan  6°. 

cot  894°  =  —  cot  6°, 

sec  894°  =  —  sec  6°, 

esc  894°  =  CSC  6°. 
(7.)  sin  267°  =  — sin  87°, 

tan  —  2540  =  —  tan  74°, 

COS  950"  =  —  cos  50°. 
(8.)  —0.28. 


ANSWERS   TO  EXERCISES 


I3» 


(9.)  2  sin'  X. 

(I9-)  53°  33'- 

(10.)   I +  560"  X. 

(20.)  1 15.136  ft. 

(II.)  sin  (;r— 90°)=  —  cos^, 

(21.)  76.355  ft. 

cos  {X  —  90=)  =  sin  -f, 

(22.)  .5  =  80°  32", 

tan  {X  —  90°)  =  —  cot  X, 

.4  =  C  =  49°  59' 44"- 

cot  (X  —  90°)  =  —  tan  X, 

(23.)       ^=53°  16' 36", 

sec  (.r  —  90=')  =  CSC  x, 

b=  12.0518  in., 

CSC  (x  —  90°)  =  —  sec  X. 

area  =  72.392  sq.  in. 

(24.)        b—  130.52  in., 

§  28  (page  24). 

area  =  24246  sq.  in. 

(25.)  23.263  ft. 

(I.)    d:  =62.324, 

(26.)  17°  48". 

A  =  32°  52'  40". 

(27.)  5.3546  in. 

(2.)     <^  =  21.874. 

(28.)  1084950  sq.  ft. 

yj  =  39°  45'  28". 

(29.)  17  ft.,  885  sq.  ft. 

^=50°  14'  32". 

(30.)       radius  =  24.882  in., 

(3.)    rt  =  300.95, 

apothem  =  20.13  in.. 

^  =  683.96, 

area=  1472  sq.  ii 

^  =  66°  15'. 

(31.)  12.861. 

(4.)       b  =  26.608, 

(32.)  1782.3  sq.  ft. 

^  =  45.763, 

(33.)  38168  ft. 

^-35°  33'. 

(34.)  20.21  ft. 

area  =  495-34- 

(35.)  2518.2  ft. 

(5-)        '^  =  3-9973. 

^  =  4.1537. 
y4  =  1 5°  46'  33", 

§  29  (page  28). 

area  =  2.257. 

(I.)  ^  =  22°  58', 

(6.)  ^  =  0.01729. 

b^y.oy. 

(7.)  rt  =  298.5. 

c  =  9.0046. 

(8.)  ^  =  39°  42' 24". 

(2.)    <^  =  79-435- 

(9.)  f  =  2346.7. 

^=45°  27'  14", 

(10.)  .^  =  28°  57' 8". 

C  =  95°  24' 46". 

(II.)  444.16  ft. 

(3.)      ^^  =  7.6745. 

(12.)  186.32  ft. 

^^'  =  2.6435, 

(1 3-)  34°  33'  44"-           ^ 

^  =  46°  43' 50". 

(14.)  303.99  ft. 

^'=133°  16'  10", 

(15.)  238.33  ft. 

ACB  =  ios°  53'  10", 

(16.)  15  miles  (about). 

ACB'  =  19°  20'  50". 

(I7-)  79.079  ft. 

(4.)  A  =  37°  53'. 

(18.)  165.68  ft. 

^  =  43°  52' 25", 

132 


ANSWERS   TO  EXERCISES 


C  =  98-  14'  35". 

(5)  902.94- 
(6.)  1253.2  ft. 
(7.)  357.224  ft. 
(8.)       A  =  44°  2'  9", 
^  =  51°  28'  II", 
C  =  843  29'  40", 
area=  126100  sq.  ft. 
(9.)  407.89  ft. 
(lO.)  B=  121°  7'  16", 
C=92°  20'  38", 
D  =  7\°  11'  6". 
(II.)  ^C  =  6.6885. 
Z?6'=  1. 991 5. 

§  34  (page  34). 

(2.)  sin  (4504- .r)  = 

^-v/2  (cos.r  +  sin^), 
COS  (45°+  ^')  = 

\  -v/2  (cosa  —  sin  jr), 
sin  (30°— jr)  = 

I  (cos.r —  -v/3  sin  x), 
cos(30"— .r)  = 

\  (^^3  cos.r  +  sin.r),     | 
sin  (6o°4-.*')  = 

i  (Vs  cos  a-  -|-sin.r), 
cos  (60°+  -r)  = 

\  (cos 4-  — -v/3  sin.i). 

(3.)  sin  (.*•+ v)=ff. 
sin  {x—y)=l^. 

„     -v/6  +  -v/2 
U.)  sin  75°=        T        • 
4 

V6  — -v/2 
4 

„       \/6  — \/2 

4 

4 


(6.)  sin(.r-|->')  =  - 


-V/15  +  A/3 


cos  (.r  —J-)  = 


3v/5  +  i 


§  39  (page  37). 

(5.)  sin  (450-^2  = 

^  ^2  (cos.t  —  sinx), 
cos  (45°  —  -r)  = 

\  \/2  (cosx  -|-  sin  x), 
sin  (45°+.r)  = 

-  ^  -v/2  (cos  .r  +  sin  .r), 
cos  (45° -(-.r)  = 

\  -v/2  (cos  X  —  sin  .1). 
(6.)  tan  75°=  2  -I--V/3. 
tan  15°  =  2  —  ^/3. 

(14.)  sini;/r-'^/3-\/5 


i/=\/' 


3+V5 


tan  \y  = 


3-V5 


cos  75°  = 
(5.)  sin  15 
cos  15 


(15.)  sin  2.r  =  — f|, 
cos  2X  =  —-^g. 

(16.)   sin  22^°  =  iy/2— -\/2. 

COS  22|°  =  ^ v/2  +  v'a, 
tan  22^°  =  v/2  —  I , 

cot  22^°  = -v/2  +  I, 

sec  22^°  =  •y/4  —  2  -v/2, 

CSC  22^°  =  W4  +  2  -v/2. 

(>7.)  -^^V- 


(18.)  sin  i5°  =  iv^2-^3, 
cos  iS°  =  h\/^-\-V3- 


ANSWERS    TO  EXERCISES 


133 


tan  i5°  =  2  — v/3, 
cot  i5=  =  2-f  Vs- 

sec  i5°  =  2-y/2  — -/s, 

CSC  I  5==  2^2  + V3- 

(20.)  sin  5-1- = 

5  sin  X  —  20  sin'  x 

-\- 16  sin*.f. 
(21.)  COS  5jr  = 

5  cos  X  —  20  cos'  X 

+  16  cos^.r. 
(23.)  The  values  oi  x  <^  360°  are 
0°,  30°,  1 50=^,  180°,  210°  330°. 
(36.)  tan.r  tan^. 

§  41  (page  40). 

(I.)  sin  — '  i -v/2  =  45°,  135°, 

45°+ 360°,  etc., 
cos— '  ^  =  60°,  300°.  etc., 
tan-'  (—  0=  135°, 315°, etc., 
cos  — •  I  =0°,  360°,  etc., 
sin  —  ■  (—  ^)  =  210°,  330°  etc, 

(2.)  tan.r  =  3. 

(3.)  cos.r  =  ±|,  tan.i-  =  ±|. 

(4.)  sin  (tan-' 1^/3)=  ±1. 

(5.)  sin(cos-'|)  =  ±f. 

(6.)  cot  (tan  -  >  j^)  =  1 7- 

(7.)  a  =  ^y/1. 

(8.)  45°,  225° 

(9.)  .^  =  45°  J  =180°. 

(10.)  sin~'rt  =  225°. 

§  48  (page  46). 

(I.)   C=I2I°  33', 
'^  =  2133-5. 

c  =  2477.8. 

(2.)  C=55°4i'. 

'^^  534-05, 


^  =  653.52. 

(3.)  C=45=34'. 

a  =  1 548. 1, 

^=:  1293.7. 

(4.)  ^  =  105=59', 

a=z  54.018, 

r  =  47.738. 
(5.)  ^  =  68°  58', 

b  —  5274.9. 

<r=:3730. 
(6.)  ^=54^^58'. 

«  =  923.4, 

c=i  1187.7. 

§  49  (page  47). 

(I.)  (I.)  Two  solutions. 

(2.)  One  solution,  a  right  tri- 
angle. 

(3.)  One  solution.. 

(4.)  Two  solutions. 
(2.)  Bz=\6°  57'  21", 

C=  15°  50' 39", 

<:=:0.32I22. 
(3.)     r=  2.5719, 

^=13°  15'  l", 

C=i42°  13' 59"- 

(4.)  f= 93.59,       ^'  =  54.069, 

^  =  26°  52' 7",  ^'=133°  7' 53", 

C  =  i3i°46'53",C'=25^3i'7". 

(5.)  No  solution. 

(6.)  ^=  1.0916,        /^'=o. 36276, 
^=39°37'i6",/?'=:i4o°22'44", 
^=ii7°5o'44",i5'=i7°5'i6". 

§  50  (page  48). 

(i.)  a  =  0.097 1, 
5  =  90°  35' 36", 

C=48^9'34". 
5  =  0.0053261. 


134 


ANSWERS    TO  EXERCISES 


(2.)    r— 14.211, 
A  =  76"  id  5", 

^  =  44°  52' 55" 

5  =  80.962. 
(3.)    ^  =  85.892, 

/I  =67°  21' 42". 
(7  =  62°  48'  18". 

6  =  3962.8. 
(4.)    a  =0.6767, 

B=\S°  9'  2\", 
C=i3i°  19'  39". 
5  =  0.08141. 
(5.)    f  =  72.87. 

A  =  40°  50'  32". 
B=\i°  2'  28". 
5  =  422.65. 

§  51  (page  49). 


(I.) 

A  = 

55° 

20' 

42". 

B  = 

106 

°  35 

'  36". 

C  = 

18" 

3'  42". 

S  = 

267.92. 

(2.)  A  = 

34° 

24' 

26". 

B  = 

73° 

14' 

56", 

C  = 

72° 

20' 

36". 

S  = 

3.61 

43- 

(3-) 

A  = 

52° 

20 

24". 

B  = 

107 

°I9 

'  14". 

C  = 

20° 

20' 

24". 

S  = 

I437.5- 

(4J 

A  = 

97° 

48' 

, 

B  = 

18° 

21 

48". 

C= 

63° 

50 

12". 

5  = 

193 

•13- 

(5.)  A  = 

=  54° 

20' 

16". 

B  = 

=  70° 

27' 

46". 

C  = 

=  54° 

72 

, 

S  = 

16090. 

(6.) 

A  = 

^35° 

59' 

30". 

^  =  48^44'  32", 

6=95^  '5' 56". 

S  =  0.60709. 

§  52  (page  50). 

(I.) 

n  16.6  ft. 

(2.)  3081.8  yards. 

(3.)  638.34  ft.. 

14653  sq.  ft. 

(4.)  4.1  and  8.1. 

(5-) 

13.27  miles. 

(6.)  6667  ft.     One  solution 

(7.) 

121.97. 

(8.) 

44°  2'  56". 

(9-) 

32.151  sq.  miles. 

(iiO 

54°  29'  12". 

(12.) 

a=  12296  ft.. 

f  =  13055  ft. 

(1 30 

294.77  ft. 

(14.) 

222.1  ft. 

(16.) 

4202.1  ft. 

(17.) 

72.613  miles. 

(18.) 

50.977  ft. 

(1 9-) 

0.85872  miles. 

(20.) 

2.98  miles. 

(21.) 

13939  ft. 

(22. 

8.2  miles. 

(23-^ 

187.39  ft. 

(24. 

0.601 1. 

(25. 

)  4.81 12  miles. 

(26. 

)  60°  51'  8". 

(27. 

)  37.365  ft. 

(28. 

)  3.2103  miles. 

(29. 

1  10.532  miles. 

(30. 

851.22  yards. 

(31. 

1  9.5722  miles. 

(32. 

)  6. 1 27 1  miles. 

(33- 

)  280.47  ft. 

(34- 

)  123.33  ft. 

ANSWERS    TO  EXERCISES 


135 


(35.)  4.81 12  miles. 
(36.)  2666.1  ft. 

§  53  (page  56). 

(I.)  30°  =  0.5236, 

45^  =  0.7854. 

60°=  1.0472, 
1 20°  =  2.0944, 
135°=  2.3562, 
720°  —  12.5664, 
990°=  17.2788. 

(2.)  J  =  22°  30', 

-=i8°, 
10 

i  =  28°  38' 53". 

1=  100°  16' 4". 

(3-)  1-35.  0.54- 

§  74  (page  73). 

(I.)  sin  4-1' =  4  cos'.r  sirur 

—  4  cos.r  sin'x, 
cos  4-ir  =  cos*x 

—  6  cos''  X  sin^  x  -\-  sin*  x. 

(2.)  sin  6jr  =  6  cos'  x  sin  x 

— 20cos';ir  sin'.r 
+  6  cos-r  sin*.r, 
cos  6x  =  cos*  X 

/        —  15  cos*  X  sin^  X 
-\-  1 5  cos"''  X  sin*  X  —  sin*  .r. 

(3.)  .r„=i,    x,  =  i  +  /-^, 

*  *  2 

X -1-/^ 


(4.).r,=  i,  .1-^  =  0. 3090+/ 0.95 1 1, 
-i",  =  —  0.8090  +  / 0.5878, 

.1'3  =  0.8090  —  /  0.5878, 

x^  =  0.3090  —  /  0.95 1  I . 

§  77  (page  78). 
(23.)  X  =  30°. 
(24.)  _y  =  30=. 
(25.)  .jr  =  0°  or  45°. 
(26.)  .r  =  6o°. 
(27.)  J  =  45^. 

(28.)  ^  =  45°- 

(29.)  .1=45°. 

(30.)  .r  =  30°. 

(31.)  ;ir  =  6o° 

(32.)  jr  =  30°. 

(33.)  No  angle  <  90°. 

(34.)  jr  =  30=. 

(35.)  sin  92°  =  cos  2°. 

(36.)  cos  127°  =  — sin  37°. 

(37.)  tan  320°  =  —  tan  40'='. 

(38.)  cot  350°  =  — cot  io=>. 

(39.)  sin  265"  =  — cos  5°. 

(40.)  tan  171°=  —tan  9=". 

(41.)  cos.r=r  — IV33, 

tan.r  =  — 5%-v/33, 

cotx  =  — iV33, 

secx  =  — jVV'33. 

CSC  X  =  J. 

(42.)  sin.ar  =  — l-v/55,  -, 

tan.r  =  i\/55. 
cotx  =  ^%\/Js. 
sec  x=  —  |, 
csc.r  =  —  A-  -\/s$. 
{43.)  sin.r  =  — W-y/^. 

COSX  =  --j%^Y^, 

cot X  =  |,  sec  X  =  —  1^  y'l  3, 


136 


ANSWERS   TO  EXERCISES 


CSC  ;jr  =  —  ^  -v/13. 
(44.)  sin.r  =  — /^-/t^. 

cos  X  =  ,\  -v/74, 

tanjr=:  — f,  sec ^=1-^/74, 

CSC  x=—\  vTi- 
(45.)  Quadrant  II  or  IV. 
(46.)  Quadrant  I  or  II. 
(47.)  Quadrant  III  or  IV. 
(48.)  Quadrant  I  or  II. 
(49.)  .r  =  o°,  I2o^  180°,  240^ 
(50.)  x  =  io°,  135°.  150°.  315°. 

(51.)  .V=:0°,   90°,    120°,    l8o^  240°, 

270°. 

(57-)  o. 

(58.)  a. 

(59.)  2{a-b). 

(60.)  ^i.a-'-b"-). 

§  78  (page  80). 
(I.)  306.32  ft. 
(2.)  831.06  ft. 
(3.)  53°  28'  14". 
(4.)  49.39  ft. 
(5.)  0.43498  mile. 
(6.)  209.53  ft. 
(7.)  7-3188  ft. 

(8.)  37°  36'  30". 

(9.)  109.28  ft. 

(10.)  502.46  ft. 

(II.)  6799.8  ft. 

(12.)  219.05  ft. 

(13.)  491.76  ft. 

(14.)  50°  32'  44". 

(15.)  49°  44'  38". 

(16.)  34-063  ft. 

(17.)  32.326  ft.,  29^  6' 35". 

(18.)  5.6569  miles  an  hour. 

(19.)  56.295  ft. 

(20.)  103.09  ft. 


(21.)  71°  33'  54". 
(22.)  858,160  miles. 
(23.)  238,850  miles. 
(24.)  2163.4  miles. 
(25.)  90,824,000  miles. 
{26.)  432.08  ft. 
(27.)  60.191  ft. 
(28.)  0.32149  mile. 
(29.)  193.77  ft. 

§  79  (page  83). 
(I.)  3.416  ft. 
(2.)  3.7865  ft. 
(3.)  20.45  ft. 
(4.)  36.024  ft. 
(5.)  8.6058  sq.  ft. 
(6.)  181.23  in. 
(7.)  2.9943  ft. 
(8.)  5. 131 1  in. 
(9.)  25.92  ft. 
(10.)  92°  i'  24". 
(II.)  1. 2491. 
(12.)  330  12' 4". 
(13.)  11248  ft. 
(14.)  0.60965  miles. 
(15.)  1.3764. 
(16.)  1.9755. 
(17.)  19.882. 
(18.)  0.9397. 
(19.)  6.4984. 
(20.)  3.4641- 
(21.)  6.1981. 
(22.)  6.9978. 
(23.)  15.25. 

§  80  (page  84). 

(78.)  X  —  90°,  1 20°,  240°,  270°. 

(79.)  x  =  o°,  20°,  45°,  90°,  100°, 
i35<=,  140°,  180°,  220°, 
225°  260°  270°,  315°, 
340°. 


ANSWERS    TO  EXERCISES 


137 


(80.)  x  =  o°,  30°,  90=^.  150°.    180°, 

270°. 
(81.)  x  =  cP,  45=,  120°,  240°,  225°, 

270°. 
(82.)  X  =  0°,  90°,  1 800,  270°. 
(83.)  ^  =  0°,  90°,  210°,  330°. 
(84.)  X  =  240°,  300°. 
(85.)  .r  =  2io°,  330°. 
(86.)  X  =  0°,  90°. 
(87.)  x  =  o°,  180='. 
(88.)  x  =  oo,  180°. 
(89.)  x  =  o°,  90°,  120°,  180°,  240=. 

270°. 
(90.)  ;r  =  45°.  135°.  225=,  315°. 
(91.)  j:  =  30°,  150°,  210°,  330°. 

§  81  (page  88). 
(I.)  2145.1  ft. 
(2.)   12.458  miles. 
(3.)  1. 1033  miles. 
(4.)   1 508.4  ft. 
(5-)  I7i9  3yards. 
(6.)   1.2564  miles. 
(7.)  1346.3  ft. 
(8.)  387.1  yards. 
(9.)  5.1083  miles. 
(10.)  3791-8  ft. 
(II.)  4.4152  ft. 
(12.)  28°  57'  20". 
(13.)  115.27. 
(14.)  44.358  ft. 
(15.)  92.258  ft. 
(16.)  101°  32'  16". 
(17.)  0.83732  mile. 
(18.)  539-1  ft. 
(19.)  1.239. 

(20.)  152.31  and  238.3. 
(21.)  68.673  ft. 
(22.)  32.071  ft. 
(23.)  13778  ft. 


(24.)  55.74  ft. 

(25.)  247.52  ft. 

(26.)  556.34  ft. 

(27.)  46572  ft. 

(28.)  109.22  ft. 

(29.)  2639.4  ft. 

(30.)  396- 54  ft. 

(31.)  287.75  ft. 

(32.)  2280.6  ft. 

(33.)  64.62  ft. 

(34.)  127.98  ft. 

(35-)  45-183  ft. 

(36.)  4365-2  ft. 

(37-)  140-17  ft. 

(38.)  610.45  ft. 

(39.)  1 56.66  ft. 

(40.)  41°  48'  39"  and  125°  25  57' 

(41.)  51,288,000. 

(42.)  366680. 

(43)  1 1 586. 

(44.)  947460. 

(45.)  0.89782. 

(46.)  9929-3- 

(47-)  751-62  sq.  ft. 

(48.)  3145-9- 
(49.)  855.1. 
(50.)  876.34. 

§  88  (page  98). 

(I.)  ^=54°  59  47". 
^  =  45°  41' 28", 

C=65°45'58". 

(2.)   C=yi-^  36'  47". 

^  =  95°  22', 
c  =  7i°  32'  14". 
(3.)  C=64°  14' 3.0". 
C'=iiS°45'3o". 
«=48^'  22'  55", 

^■  =  131°  37   5". 
^=42"  19  17". 


138 


ANSWERS    TO  EXERCISES 


r  =  i37''4o'43". 
(4.)  C  =  65°49'  54". 

a  =  63°  10'  6". 

^  =  38°  59   12". 
(5.)  rt  =  75°  13'  i", 

7^=58°  25' 46", 

C  =  67°  27'  i". 
(6.)  a  =  76°  30'  37", 

d^es"  28'  58," 

f  =  55°  47' 44". 
(7.)  />'  =  54°44'23". 

/J  =  64°  36' 39". 

r  =  47°  57' 45"- 
(8.)  B  =  g6°  13'  23". 

f?  =  73°  17'  29", 

c  =  70°  8'  38". 
(9.)  B  =  66°  58', 

«=  11°  35' 49". 
^  =  4°  35' 26". 
(10.)  ^  =  61°  4'  55". 
^  =  40°  30'  22", 
c  =  50°  30'  32". 

§  99  (page  107). 

(I.)  f  =  155°  35'  22", 
75=10°  19' 34". 
C=  171''  48'  22". 

(2.)  rt  =  i3i°36'36". 
^=116°  36'  38", 
f  =  29°  II'  42". 

(3.)  «  =  107°  7' 45". 
^  =  48°  57' 29". 
C  =  62°  31'  40". 

(4.)  ^  =  62°  54' 43", 
a=  114°  30'  26", 
f  =  56°39'io". 


(5-)^  = 

=  130 

°  35'  56". 

B  = 

=  30" 

25'  34". 

•  C= 

=  31° 

26'  32". 

(6.)  «  = 

■98° 

21'  22", 

d  = 

109 

'  50'  8", 

c  = 

:II5 

°I3'4". 

(7.)  ^  = 

=  32° 

26'  9". 

a  — 

:84° 

14'  32". 

c  = 

51° 

6'  12". 

(8.)  «  = 

:8o° 

5'  8". 

b  = 

70° 

10'  36", 

c  = 

H5 

=  5'  2". 

(9.)  ^  = 

70° 

39' 4". 

B- 

48° 

36'  2", 

C  = 

119 

'  15' 2". 

{10.)  a  = 

40° 

0'  12". 

B- 

42° 

15'  u". 

C  = 

121' 

'  36'  19". 

§  100  (page  109). 

(I.)  80.895  sq.  in. 

(2.)  26.869  sq.  in. 

(3,)  158.41  sq.  in. 

(4-)  39990  sq.  miles. 

§  loi  (page  112). 

(I.)  5C  =  48°2'43". 
^C=52°  53' 9". 
(2.)  7  :  24  A.M. 

(3.)  4  P.M. 

§  102  (page  114). 

(I.)  3029t\j  miles. 
(2.)  2229.8  miles. 
(3.)  2748.5  miles. 
(4.)  7516.3  miles. 
(5,)  5108.9  miles. 


THE   END 


LOGARITHMIC 


AND 


TRIGONOMETRIC  TABLES 


FIVE- PLACE  AND  FOUR-PLACE 


PHILLIPS-LOOMIS  MATHEMATICAL   SERIES 


LOGARITHMIC 


AND 


TRIGONOMETRIC  TABLES 


FIVE-PLACE  AND  FOUR-PLACE 


BY 

ANDREW    W.   PHILLIPS,   Ph.D. 

AND 

WENDELL  M.  STRONG,  Ph.D. 

YALE  UNIVERSITY 


NEW   YORK   AND    LONDON 

HARPER    &     BROTHERS    PUBLISHERS 

1899 


THE  PHILLIPS-LOOMIS  MATHEMATICAL  SERIES. 

ELEMENTS  OF  GEOMETRY.     By  Andrew  W.  Phillips,  Ph.D., 

and  Irving  Fisher,  Ph.D.    Crown  8vo,  Half  Leather,  $1  75.    [By 

mail,  $1  92.] 
ABRIDGED  GEOMETRY.     By  Andrew  W.  Phillips,  Ph.D.,  and 

Irving    Fisher,   Ph.D.      Crown   8vo,   Half   Leather,  $1    25.      [By 

mail,  $1  40.] 

PLANE  GEOMETRY.  By  Andrew  W.  Phillips,  Ph.D.,  and  Irving 
Fisher,  Ph.D.    Crown  8vo,  Cloth,  80  cents.    [By  mail,  90  cents.] 

GEOMETRY  OF  SPACE.  By  Andrew  \V.  Phillips,  Ph.D.,  and 
Irving  Fisher,  Ph.D.    Crown  8vo,  Cloth,  $1  25.    [By  mail,  $1  35.] 

OBSERVATIONAL  GEOMETRY.  By  William  T.  Campbell,  A.M. 
Crown  8vo,  Cloth. 

ELEMENTS  OF  TRIGONOMETRY,  Plane  and  Spherical.  By 
Andrew  VV.  Phillips,  Ph.D.,  and  Wendell  M.  Strong,  Ph.D.,  Yale 
University.    Crown  8vo,  Cloth,  90  cents.    [By  mail,  98  cents.] 

LOGARITHMIC  AND  TRIGONOMETRIC  TABLES.  Five-Place 
and  Four- Place.  By  Andrew  W.  Phillips,  Ph.D.,  and  Wendell 
M.  Strong,  Ph.D.    Crown  8vo,  Cloth,  $1  00.    [By  mail,  $1  08.] 

TRIGONOMETRY  AND  TABLES.  By  Andrew  W.  Phillips, 
Ph.D.,  and  Wendell  M.  Strong,  Ph.D.  In  One  Volume.  Crown 
8vo,  Half  Leather,  $1  40.    [By  mail,  $1  54.] 

LOGARITHMS  OF  NUMBERS.  Five-Figure  Table  to  Accompany 
the  "  Elements  of  Geometry,"  by  Andrew  W.  Phillips,  Ph.D.,  and 
Irving  Fisher,  Ph.D.  Crown8vo,  Cloth,  30  cents.  [By  mail,  35  cents.] 


NEW   YORK   AND   LONDON  : 
HARPER   &    BROTHERS,  PUBLISHERS. 


Copyright,  1898,  by  Harper  &  Brothers 

Alt  rights  restrved. 


CONTENTS 


TABLB  PAGH 

Introduction  to  the  Tables v 

I.  Five-place  Logarithms  of  Numbers i 

II.  Five -PLACE    Logarithms    of   the   Trigonometric 

Functions  to  Every  Minute 29 

III.  Five-place  Logarithms  of  the  Sine  and  Tangent  _^ 

OF  Small  Angles 121 

IV.  Four-place  Naperian  Logarithms 131 

V.  Four-place  Logarithms  of  Numbers 135 

VI.  Four -PLACE    Logarithms    of  the  Trigonometric 

Functions  to  Every  Ten  Minutes 139 

VII.  Four -PLACE  Natural  Trigonometric   Functions 

TO  Every  Ten  Minutes 149 

VIIL  Squares  and  Square  Roots  of  Numbers     ....  159 
IX.  The  Hyperbolic  and  Exponential  Functions  of 

Numbers  from  o  to  2.5  at  Intervals  of  .1  .    .  160 
X.  Constants  —  Measures  and   Weights  and  Other 

Constants 161 


INTRODUCTION  TO  THE  TABLES 


COMMON    LOGARITHMS. 

1,  The  common  logarithm  of  a  number  is  the  index  of 
the  power  to  which  lo  must  be  raised  to  give  the  number. 

Thus,             log  loo    =  2,          because  loo    =  lo'' 

log      I     -=0,               "  I    =10° 

log        .1=  — I,           "  .1  =  10-' 

log      3     =47712,        "  3     ^lO"*""' 

In  general,  \ogm  =  x  H  mj=  10*. 

2,  To  multiply  two  numbers,  add  their  logarithms.  The 
result  is  the  logarithm  of  the  product. 

Proof. —  Um  =  icy    so  that  log  ;«   =:x, 

and  n  =  io>'      *'      "     log  n     =y, 

then  ;««=io*+-''"      "     logmn  =  x~\-y. 

Hence  log  mft  =  log;«  +  log  n. 

3,  To  divide  one  number  by  another,  subtract  the  loga- 
rithm of  the  divisor  from  the  logarithm  of  the  dividend. 
The  result  is  the  logarithm  of  the  quotient. 

„      ,  m     10* 

Proof—  -  =  --  =  lO'-y ; 

Hence  \og~=x—y  =  \ogm  —  \ogn. 

n 

4,  To  raise  a  number  to  a  power ^  multiply  the  logaritJun 
of  the  number  by  the  index  of  the  power.  The  result  is  the 
logarithm  of  the  power. 


vi  INTRODUCTION   TO    THE    TABLES. 

Proof.  —  m'     =  ( I  o-*") '  =  I  o'" ; 

H ence  log  m"  =  ax  =  a  log  m. 

S*  To  extract  a  root  of  a  number,  divide  the  logarithm  of 
the  number  by  the  index  of  the  root.  The  result  is  the  loga- 
rithm of  the  root. 

* 

Proof. —  ^  /;//  —  *  /lo*  =  TO*. 

-.  ,      *   /         X      log;« 

Hence  log  ^  /  in  =  -=  -  ,-  • 

6.  Restatement  of  laws  : 

log  nm  =  log  tn  +  log  n ; 

log —  =  log»n-logw; 

log  »w"  =  a  log  »n ; 
*  /         logm 


7.  Most  numbers  are  not  integral  powers  of  lo;   hence 
most  logarithms  are  of  decimal  form. 

Thus,  log  2. 2  — .34242,    log4  =  . 60206. 

8,  If  a  logarithm  is  negative,  it  is  expressed  for  conven- 
ience as  a  negative  integer  plus  2i  positive  decimal. 

The  logarithm  of  a  number  less  than  i  is  negative. 
The  negative  integer  is  usually  expressed   in  the  form 
9—10,  8—  10,  etc. 

Thus,  log. 2 1 544  =— I +  .33333.  written  9.33333— 10; 
log  .021 544  =  —  2+. 33333.  "  8.33333—10; 
log  .002 1 544  =  —  3 +.33333,       "       7-33333— 10. 

Remark. — In  some  books  the  negfative  integer  is  written  i,  2,  etc., 
instead  of  9  —  10,  8—10,  etc. 

The  integral  part  of  a  logarithm  is  the  characteristic; 
the  decimal  part  is  the  mantissa. 

Thus,  log 2 1 5.44  =  2.33333;  ^he  characteristic  is  +2;  the  mantissa 


COMMON  LOGARITHMS.  vii 

is  4--33333:   log  .021544=8.33333— lo;    the  characteristic  is  8  —  10 
=  —  2  ;  the  mantissa  is  +  •33333- 

f>.  It  is  evident  that  the  larger  a  number  the  larger  its  logarithm. 
Hence  the  logarithm  of  any  number 

between    i       and    10      is      o+a  mantissa, 
.<         jQ        ..     jQQ       ..       I  -|- "         " 

.1      "        I        "-I  +  " 
.01    "  .1     "  —  2  +  "         "        etc. 

We  have,  then,  the  following  rule  for  obtaining  the  characteristic; 

10,  Count  the  number  of  places  the  first  left-hand  digit  of 
the  number  is  removed  from  the  unit's  place. 

If  this  digit  is  to  the  left  of  the  unit' s  place  ^  the  result  is  the 
required  characteristic. 

If  this  digit  is  to  the  right  of  the  unifs  place,  the  result 
taken  with  a  minus  sign  is  the  required  characteristic. 

If  this  digit  is  in  the  unit's  place,  the  characteristic  is  zero. 
Thus  the  characteristic  of  the  logarithm  of  21550  is      4 

"     "  "  ".       21.55         "       I 

2.155       "       o 

' "  -2155    "—I 

"     "  "  "  -02155  "  —  2 

11,  The  logarithms  of  numbers  which  differ  only  in  the 
position  of  the  decimal  point  have  the  same  mantissa. 

For  to  change  the  position  of  the  decimal  point  is  to  multiply  or 
divide  by  an  integral  power  of  10;  that  is,  an  integer  is  added  to  or 
subtracted  from  the  logarithm,  and  consequently  only  the  character- 
istic is  changed. 

Thus,  log  2 1 544  =3-33333 

log       2.1544     =0.33333 
log         .21544   =9-33333—10 
log        .021544  =  8.33333—10 
Therefore,  in  finding  the  mantissa  of  the  logarithm  of  a 
number  the  decimal  point  may  be  disregarded.     The  man- 
tissa is  found  from  the  tables  of  losfarithms. 


viii  INTRODUCTION  TO    THE    TABLES. 

USE   OF  THE  TABLE  OF  LOGARITHMS  OF   NUMBERS. 

(table  I.) 

12,   To  fi7td  the  logarithm  of  a  Clumber. 

Look  in  the  column  at  the  head  of  which  is  "  N  "  for  the 
first  three  figures  of  the  number,  and  in  the  line  with  "N,"  for 
the  fourth  figure.  In  the  line  opposite  the  first  three  figures 
and  in  the  column  under  the  fourth  is  the  desired  mantissa. 

Only  the  last  three  figures  of  the  mantissa  are  found  thus ;  the 
first  two  must  be  taken  from  the  first  column  ;  they  are  found  either 
in  the  same  line  or  in  the  first  line  above  which  gives  the  whole  man- 
tissa, except  when  a  *  occurs.  If  a  *  precedes  the  last  three  figures  of 
the  mantissa  the  first  two  are  found  in  the  following  line  : 

The  characteristic  is  obtained  by  §  lO. 

Exatnple. — To  find  the  logarithm  of  105400. 

The  characteristic  =  5.  §  10 

The  mantissa         =   .02284  (opposite  105  and  under  4  in  the  tables) ; 

Hence  log  105400  =  5.02284. 

IS,  If  there  are  five  or  more  figures  in  a  number  the 
figures  beyond  the  fourth  are  treated  as  a  decimal.  The 
corresponding  mantissa  is  between  two  successive  mantissas 
of  the  tables. 

Example. — To  find  the  logarithm  of  10543. 

The  characteristic  =  4.  §10 

The  mantissa  is  not  in  the  tables,  but  is  between  the  mantissa  of 
1055  =  .02325  1 

and  the  mantissa  of  1054  =  .02284 

Their  difference  =        41 

Hence  an  increase  of  one  in  the  fourth  figure  of  the  number  pro- 
duces an  increase  of  41  in  the  mantissa.  Then  an  increase  of  .3  must 
produce  an  increase  of  41  X  .3  in  the  mantissa. 

41  X. 3=  12.3=  12  nearly. 

Hence  the  mantissa  of  10543  =  .022844- 12  =  .02296. 

Therefore  log  10543=  4.02296. 


LOGARITHMS  OF  NUMBERS.  ix 

An  easy  method  of  multiplying  41  by  .3  is  to  use  the  table  of  pro- 
portional parts  at  the  bottom  of  the  page  in  the  tables. 
Under  41  and  opposite  3  is  I2.3(=4i  X.3). 

14z,  Figures  beyond  the  fifth  are  usually  omitted  in  the 
use  of  a  five -place  table,  as  their  retention  does  not  add 
much  to  the  accuracy  of  the  result.  For  the  fifth  figure, 
however,  we  choose  the  one  which  gives  most  nearly  the 
true  value  of  the  number. 

Thus,  if  the  number  is  157.032,  we  use  157.03; 
"  "  "  "  157.036,  "  "  157.04; 
"    "         "        "   157.035.   "     "     157-04. 

15,   To  find  a  number  from  its  logarithm. 
The  process  is  the  reverse  of  finding  the  logarithm  from 
the  number ;   it  is  illustrated  by  the  following  examples : 
Find  the  number  of  which  9.12872  — 10  is  the  logarithm. 
Since  the  characteristic  =  —  i,  the  decimal  point  will  be  before  the 
first  figure  of  the  number. 

.12872  is  opposite  134  and  under  5  in  the  tables. 
Hence  .12872  =  the  mantissa  of  1345, 

and  9.12872— io  =  log. 1345. 

Find  the  number  of  which  9.12895  —  10  is  the  logarithm. 
The  mantissa  .12895  is  not  in  the  tables,  but  is 
between  .12905  =  mantissa  of  1346 

and  .12872=        "         "  1345. 

.00033  =  the  diflference. 
.  1 2895  =1  mantissa  given, 

.12872  =  mantissa  of  1345,  the  smaller  number, 
23  =  the  difference. 
Change  ff  into  a  decimal.    The  first  figfure  of  this  decimal  will  be 
the  figure  in  the  fifth  place  of  the  number. 

II  =  .7  nearly. 
Hence  9.12895  — io  =  log. 13457. 


X  INTRODUCTION   TO    THE  TABLES. 

An  easy  method  of  changing  §^  into  a  decimal  is  to  use  the  table 
of  proportional  parts. 

Under  33  is  found  23.1  (=  23  nearly),  which  is  opposite  7. 

Hence  M  =  -7  nearly. 

The  process  we  have  employed  in  finding  the  logarithm 
of  a  number  of  more  than  four  figures,  or  the  number  corre- 
sponding to  a  mantissa  not  given  in  the  table,  is  called  in- 
terpolation. 

EXAMPLES  FOR  THE   USE   OF  LOGARITHMS. 

16,  Multiply  5789.2  by  .018315. 

log  5789.2  =  3.76262 
log.01831 5  =8.26281  —  10 

2.02543  =  log  106.03 
Multiply  9.8764  by  .10013. 

log  9.8764  ■=  0.99460 
log. 10013  =  9.00056—  10 

"9.99516  — 10  =  log  .98892 
Find  the  value  of  3. 141 6  x  7638.6  x  .017829. 
log  3.1416  =  0.49715 
log  7638.6  =  3.88302 
log  .01 7829  =  8.251 13—  10 

2.631 30  =  log  427.86 
Divide  81.321  by  3. 14 16, 

log8i.3i2  =  1.91021 
log3.i4i6  =  o.497i5 

1.41306  =log25.886 
Find  the  value  of  (2.1345)*. 

log  2. 1 345  =0.32930 

5 

1.64650  =  log  44.310 
Find  the  value  ofv'.oio2i. 

log  .01021  =   8.00903  —  10 
=  28.00903  —  30 

28.00903  —  30 

=9.33634— 10  =  log  .2 1694 


LOGARITHMS  OF   TRIGONOMETRIC  FUNCTIONS,     xi 

i7.  The  logarithm  of  —  is  called  the  cologarithm  of  ;;/, 

and  is  obtained  by  subtracting  log  m  from  zero. 

Thus,  if  log  w  =  9.76423—  10,  colog;«  =  0.23577. 

It  is  frequently  shorter  to  add  cologw  than  to  subtract 
logw  when  we  wish  to  divide  by  a  number  m. 

The  following  example  illustrates  this: 

Find  the  value  of  57fX  42-^4. 
644.32 

log  57.98  =1.76328 

log  42. 24=  1.62572 

colog  644.32  =  7.19090— 10 

0.57990  =  log  3.801 

USE   OF  THE   TABLE   OF  LOGARITHMS   OF   TRIGONOMETRIC 
FUNCTIONS,      (table   II.) 

18,  For  an  angle  less  than  45°,  the  degrees  are  at  the 
head  of  the  page,  the  minutes  in  the  column  at  the  left,  and 
"  L.  Sin.,"  "L.  Tang.,"  etc.,  at  the  head  of  the  correspond- 
ing columns.  For  angles  between  45°  and  90°,  the  degrees 
are  at  \\\q.  foot  of  the  page,  the  minutes  in  the  column  at 
the  right,  and  "  L.  Sin.,"  "  L.  Tang.,"  etc.,  at  the  foot  of  the 
corresponding  columns. 

The  characteristic  is  printed  10  too  large  where  it  would 
otherwise  be  negative.  Hence,  in  using  this  table,  — 10  is 
to  be  supplied,  except  for  the  cotangent  of  angles  less  than 
45°  and  the  tangent  of  angles  from  45°  to  90°. 

EXAMPLES. 

log  sin  15°  25' =  9.42461  — ro. 
log  tan  28°  1 7' =  9.73084— 10. 
log  cos  62°  14'  =  9.66827  — 10. 
log  cot  25°  34' =  0.32020. 


xii  INTRODUCTION    TO    THE   TABLES. 

19.  If  the  given  angle  contains  seconds,  we  may  reduce 
the  seconds  to  a  decimal  of  a  minute  and  proceed  as  in 
finding  the  logarithms  of  numbers.  It  must  be  remem- 
bered, however,  that  log  cos  and  log  cot  decrease  as  the 
angle  increases. 

In  practice  we  remember  that  6"  is  one-tenth  of  a  minute,  ^nd  di- 
vide the  number  of  seconds  by  6",  then  use  the  table  of  proportional 
parts  at  the  bottom  of  the  page. 

EXAMPLES. 
Find  log  sin  28°  14'  36"  (=log  sin  28°  14.6'). 

log  sin  28°  15'  — log  sin  28°  14' =  23  (found  in  column  "d.") 
log  sin  28°  14' =  9.67492  — 10 
23  X. 6=  13.8=  14  nearly 

log  sin  28°  14'  36"  =  9.67506— 10 

Find  log  cos  39°  17'  22"  (=log  cos  39°  17.3^'). 
log  cos  39°  1 7' =  9.8887  5  — 10 

IOX.3f=:  4 

log  cos  39°  17'  22"  =  9.8887i  —  10 

Find  log  tan  51°  27'  44"  ( =log  tan  51°  27. 7^). 
log  tan  51°  27 '  =  .09862 

26x.7i=_    19 
log  tan  51°  27'  44"  =  .09881 

i 

Find  log  cot  67°  18'  46". 

log  cot  67°  18'  =9.62150  — 10 
36  X  .7^  =  28 

Hence  log  cot  67°  18'  46"  =  9.62 122  — 10 

20,  The  process  of  finding  an  angle,  if  its  logarithmic 
sine  or  tangent,  etc.,  is  given,  is  the  reverse  of  the  pre- 
ceding. 


EXPLANATION  OF   THE    TABLES.  xiii 

EXAMPLES. 
Given  log  sin  jr  =  9.67433  —  lo ;  find  x. 

log  sin  28°  II' =9.6742 1  —  10 
log  sin  ;r  —  log  sin  28°  11' r=  12 

and         log  sin  28°  12'  — log  sin  28°  II' =  24 

Hence  x  =  28°  1 1'  30"  (^  of  i'  being  30' ). 

Find  the  angle  whose  log  cos  =  9.88231  —  10. 
log  cos  40°  18' =  9.88234— 10. 
6o"x^=i6". 
Hence  log  cos  40°  18'  16"  =  9.88231  — 10. 

Find  the  angle  whose  log  tan  =0.17844, 
log  tan  56°  27  =0.17839. 
6o"X^=ii". 
Hence  log  tan  56"^  27'  ii"  =  o. 17844, 

Find  the  angle  whose  log  cot  =  9.87432 — 10. 
log  cot  53<3  10' =  9.87448  — 10. 

6o"X^  =  37". 
Hence  log  cot  53°  10'  37"  =  9.87432—  10. 

EXPLANATION  OF  THE  TABLES. 

21.  A  dash  above  the  terminal  5  of  a  mantissa,  as  5,  de- 
notes that  the  true  value  is  less  than  5. 

Thus,  log  389  =  2.5899496  to  seven  places,  but  to  five  placas 
log  389  =  2.58995. 

Tables  I  and  II  have  already  been  explained. 

TABLE  III. 

22.  The  logarithmic  sine  and  tangent  cannot  be  obtained 
very  accurately  from  Table  II  if  the  angle  contains  seconds 
and  is  less  than  2°. 

Table  III  is  to  be  used  when  greater  accuracy  in  the  sine 
or  tangent  of  a  small  angle  is  desired  than  can  be  obtained 


xiv  INTRODUCTION   TO    THE   TABLES. 

by  the  use  of  Table  II.  It  is  to  be  noted  that  the  first  page 
of  Table  III  gives  the  sine  and  tangent  to  every  second  for 
angles  less  than  8'. 

TABLE   IV. 

23,  Naperian  or  "  natural "  logarithms  are  logarithms  to 
the  base  e  (=2.71828  +  ).  The  whole  logarithm  is  given, 
since  the  integral  part  cannot  be  supplied  by  inspection,  as 
with  common  logarithms. 

TABLES   V   AND   VL 

24:.  Four-place  logarithms  and  logarithmic  functions  are 
used  instead  of  five-place  if  the  results  are  sufficiently  ac- 
curate for  the  purpose  in  view. 

In  Table  VI  both  the  degrees  and  minutes  are  in  the  col- 
umns at  the  sides  of  the  page,  otherwise  this  table  does  not 
differ  in  form  from  Table  II. 

TABLE  VIL 
25,  This  table  is  identical  with  Table  VI   in  form,  but 
gives  the   trigonometric   functions   themselves,  instead   of 
their  logarithms. 

TABLES   VIII,    IX,   X. 

2Q,  These  tables  require  no  explanation. 


TABLE  I 

FIVE -PLACE     LOGARITHMS 
OF     NUMBERS 


100-130 


N 

0 

1 

2 

a 

4 

5 

0 

7 

8 

9 

100 

lOI 

00  000 

043 

087 

i3o 

173 

217 

260 

3o3 

346 

389 

432 

475 

5i8 

56i 

6o4 

647 

689 

732 

775 

817 

1 02 

860 

903 

945 

988 

*o3o 

♦072 

"115 

*i57 

*'99 

*242 

io3 

01 284 

«326 

368 

4io 

452 

494 

536 

578 

620 

662 

io4 

703 

745 

787 

828 

870 

912 

953 

995 

*o36 

*o78 

io5 

02  I  19 

160 

202 

243 

284 

325 

36G 

407 

449 

490 

106 

53i 

572 

612 

653 

1^4 

735 

776 

816 

857 

898 

107 

938 

979 

*oi9 

*o6o 

*IOO 

*i4i 

<'i8i 

*222 

♦262 

*302 

108 

o3  342 

383 

423 

463 

5o3 

543 

583 

623 

663 

703 

109 
110 

III 

743 

782 

822 

862 

902 

941 

981 

*021 

*o6o 

1 

*IOO 

o4  1 39 

179 

218 

258 

297 

336 

376 

415 

454 

493 

532 

571 

610 

650 

689 

727 

766 

8o5 

844 

883 

112 

922 

961 

999 

*o38 

*o77 

*ii5 

*i54 

♦192 

*23l 

♦269 

ii3 

o5  3o8 

346 

385 

4'23 

46 1 

500 

538 

576 

6i4 

652 

ii4 

690 

729 

767 

805 

843 

881 

918 

956 

994 

*032 

ii5 

06  070 

108 

i45 

i83 

221 

258 

296 

333 

371 

408 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

117 

819 

856 

893 

930 

967 

*oo4 

*o4i 

♦078 

*ii5 

*i5i 

118 

07  188 

225 

262 

298 

335 

372 

4o8 

445 

482 

5i8 

119 
120 

121 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

918 

954 

990 

*027 

*o63 

*o99 

*i35 

*i7i 

*207 

*243 

08  279 

3 

i4 

35o 

386 

422 

458 

493 

529 

565 

600 

122 

636 

672 

707 

743 

778 

8i4 

849 

884 

920 

955 

123 

991 

*026 

*o6i 

♦096 

*l32 

*i67 

*202 

*237 

♦272 

*3o7 

124 

09  342 

377 

4l2 

447 

482 

5t7 

552 

587 

621 

656 

125 

691 

726 

760 

795 

83o 

864 

899 

934 

968 

*oo3 

126 

10  037 

072 

106 

i4o 

'75 

209 

243 

278 

3l2 

346 

127 

38o 

415 

449 

483 

5i7 

55i 

585 

6.9 

653 

687 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

*025 

129 
130 

1 1  059 

093 

126 

160 

193 

227 

261 

294 

327 

36f 

394 

428 

46 1 

494 

528 

56i 

594 

628 

661 

694 

N 

0 

1 

2 

3 

4 

5 

a 

7 

8  1  i>  1 

PP  44   43    4 

2 

41    40    39      38    37    36  | 

I 

4.4 

4.3 

4 

2 

I 

4.1 

4.0 

3.9 

I 

3.8 

3.7 

3.6 

2 

8.8 

8.6 

8 

4 

2 

8.2 

8.0 

7.8 

2 

7.6 

7-4 

7.2 

3 

l3.2 

12.9 

12 

6 

3 

12.3 

12.0 

1 1.7 

3 

11.4 

I  I.I 

10.8 

4 

17.6 

17.2 

16 

8 

4 

16.4 

16.0 

i5.6 

4 

l5.2 

i4.8 

i4.4 

5 

22.0 

21.5 

21 

0 

5 

20.5 

20.0 

19.5 

5 

19.0 

18.5 

18.0 

6 

26.4 

25.8 

25 

.2 

6 

24.6 

24.0 

23.4 

6 

22.8 

22.2 

21.6 

7 

3o.8 

3o.i 

29 

.4 

7 

28.7 

28.0 

27.3 

7 

26.6 

25.9 

25.2 

8 

35,2 

34.4 

33 

.6 

8 

32.8 

32.0 

3l.2 

8 

3o.4 

29.6 

28.8 

9 

39.6 

38.7 

Jl 

.8 

± 

36.9 

36. 0 

35.1 

-1- 

34.2 

33  3 

32.4 

130-160 


k 

0 

1 

2 

3 

4  (  5 

6 

7 

8  1  9  1 

IfO 

11  394 

428 

46i 

494 

528 

56i 

594 

628 

661 

694 

i33 

727 

»2  o57 

385 

760 
ogo 

4i8 

793 

123 

45o 

826 
i56 

483 

860 
189 
5i6 

893 
222 

548 

926 
2.54 
58 1 

959 

287 
6i3 

992 

320 

646 

*024 

352 

678 

1 34 
i35 
i36 

710 

i3o33 

354 

743 
066 
386 

775 

098 
4i8 

808 
i3o 
450 

84o 
162 
48 1 

872 
194 
5i3 

905 
226 
545 

937 
258 
577 

969 
290 
609 

*OOI 
322 

64o 

i37 
i38 
1 39 

140 

i4r 
142 
143 

672 

988 
i4  3oi 

704 

"019 

333 

735 

*o5i 

364 

767 

*o82 

395 

799 

*ri4 

426 

83o 

*i45 

457 

862 

*I76 

489 

893 

*208 
520 

925 

*239 

55i 

956 

*270 

582 

6i3 

644 

675 

706 

737 

768 

799 

829 

860 

891 

922 

i5  229 

534 

953 
269 
564 

983 
290 
594 

*oi4 
320 
625 

*o45 
35i 
655 

*o76 
38i 
685 

*io6 

4l2 

7.5 

*i37 
442 
746 

*i68 
473 
776 

*i98 
5o3 
806 

144 
i45 
i46 

836 

16  i37 

435 

866 
167 
4G5 

897 
197 
495 

927 
227 

524 

957 
256 
554 

987 
286 
584 

*oi7 
3i6 
6i3 

*o47 
346 
643 

♦077 
376 
673 

*io7 
4o6 
702 

i47 
1 48 
149 

150 

i5i 

I  52 

i53 

732 

17  026 

319 

761 
o56 

348 

791 
085 
377 

820 
114 
4o6 

850 
i43 
435 

879 
173 

464 

909 

202 
493 

938 

23l 
522 

967 
260 
55i 

997 
289 
58o 

609 

638 

667 

696 

7^5 

754 

782 

811 

84o 

869 

898 

18  i84 

469 

926 

2l3 

498 

955 
241 
526 

9B4 
270 
554 

*oi3 
298 
583 

*o4i 
327 
611 

''070 
355 
639 

*099 
384 
667 

*I27 
4l2 
696 

*i56 
44 1 
724 

1 54 
i55 
i56 

752 
19  o33 

3l2 

780 
061 
340 

808 
089 
368 

837 
117 
396 

865 
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160 

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358 

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385 

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520 

548 

575 

602 

629 

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1  1  2 

S 

4 

5    6 

7 

8  19  1 

PP  35    34    83 

32 

31    30      29    28    27  |j 

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2 
3 

3.5 

7.0 

10.5 

3.4 

6.8 

10.2 

3.3 
6.6 
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2 
3 

3.3 

6.4 
9.6 

3.1 
6.2 
9.3 

3.(. 
6.0 
9.0 

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2 
3 

2.9 

5.8 
8.7 

2.8 
5.6 
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2.7 
5.4 
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4 
5 

6 

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17.5 
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16.5 
19.8 

4 
5 
6 

12.8 
16.0 
19.2  , 

12.4 
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12.0 
1 5.0 
18.0 

4 
5 
6 

11.6 
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17.4 

1 1. 2 
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16.8 

10.8 
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16.2 

7 

8 

24.5 
28.0 
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23.8 
27.2 
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23.1 

26.4 
29.7 

7 

8 

9 

22.4 
25.6 

28.8 

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24.8 
27.9 

21.0 
24.0 

27.0 

7 
8 

9 

20.3 

23.2 
26.1 

19.6 
22.4 

25.2 

18.9 
21.6 
24.3 

160— 190 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

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2  0  4 1 2 

439 

466 

493 

520 

548 

575 

602 

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710 

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763 

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817 

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871 

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925 

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21  219 

245 

272 

299 

325 

352 

378 

405 

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484 

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537 

564 

590 

617 

643 

669 

696 

723 

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801 

827 

854 

880 

906 

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166 

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037 

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167 

194 

220 

246 

167 

272 

298 

324 

350 

376 

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169 
170 

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198 

223 

249 

274 

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325 

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426 

452 

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502 

528 

172 

553 

578 

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654 

679 

704 

729 

754 

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17.3 

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855 

880 

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930 

955 

980 

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174 

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080 

105 

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180 

204 

229 

254 

279 

175 

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353 

378 

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452 

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176 

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576 

601 

625 

650 

674 

699 

724 

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773 

177 

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871 

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920 

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969 

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2  5  o42 

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139 

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188 

2  1  2 

237 

261 

179 
180 

181 

285 

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334 

358 

382 

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455 

479 

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575 

660 

624 

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672 

696 

720 

744 

768 

792 

816 

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864 

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912 

935 

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102 

126 

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174 

198 

221 

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245 

269 

293 

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364 

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553 

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600 

623 

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670 

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881 

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186 

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27  i84 

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189 
190 

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875 

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3 

4 

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20 

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2.7 

2.6 

2 

5 

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2.4 

2.3 

2.2 

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2.1 

2.0 

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2 

54 

5.2 

5 

0 

2 

4.8 

4.6 

4.4 

2 

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4.0 

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7.8 

7 

5 

3 

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6.9 

6.6 

3 

6.3 

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0 

4 

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5 

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5 

10.5 

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6 

16.2 

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0 

6 

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6 

12.6 

12.0 

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7 

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7 

16.8 

16.1 

1 5.4 

7 

14.7 

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i3.3 

8 

21.6 

20.8 

20 

0 

B 

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18.4 

17.6 

8 

16.8 

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9 

24.3 

23.4 

22 

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2, 

21.6 

20.7 

19.8 

9 

18.9 

18.0 

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190— 230 

N 

0 

1 

2 

3   4 

5 

6 

7 

8 

9 

190 

191 
192 
193 

27  875 

898 

921 

944 

967 

989 

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126 
353 
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149 
375 
601 

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421 
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217 
443 
668 

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466 
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262 
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285 
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307 
533 

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194 

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196 

780 

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226 

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248 

825 
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270 

847 
070 
292 

870 
092 
3i4 

892 

I  I  A 

336 

914 

137 
358 

937 
159 
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959 
181 
4o3 

981 

203 

425 

197 
198 
199 

200 

201 
202 
2o3 

447 
667 
885 

469 

688 
907 

491 
710 
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732 
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535 
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973 

557 
776 
994 

579 

798 

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601 

820 
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623 

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645 

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125 

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168 

190 

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233 

255 

276 

298 

320 

535 
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557 
771 

363 
578 
792 

384 
600 
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835 

428 
643 
856 

449 
664 
878 

471 
685 
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492 
707 
920 

5i4 
728 
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204 
205 

206 

963 
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387 

984 
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218 
429 

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239 
450 

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260 
471 

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281 
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323 
534 

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345 
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366 
576 

207 

208 

209 
210 

211 
212 

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597 

806 

32  org 

618 
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639 

848 
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660 
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077 

681 
890 
098 

702 
911 
118 

723 
931 
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744 
952 
160 

765 
973 
181 

785 

994 
201 

222 

243 

263 

284 

305 

325 

346 

366 

387 

4o8 

428 
634 
838 

449 
654 
858 

469 

675 
879 

490 
695 

899 

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919 

53i 
736 
940 

552 
756 
960 

572 

777 
980 

593 
797 

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216 

33  o4i 

244 
445 

062 
264 
465 

082 

284 
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102 

3o4 
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122 
325 
526 

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345 
546 

1 63 
365 
566 

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385 
586 

2o3 

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606 

224 
425 
626 

217 
218 

219 
220 

221 
222 

223 

646 

846 

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666 
866 
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686 
885 
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706 
905 
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726 
925 
124 

746 
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766 
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786 

985 
1 83 

806 

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826 

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223 

242 

262 

282 

3oi 

321 

34i 

36i 

38o 

4oo 

420 

439 
635 
83o 

459 

655 
850 

479 
674 
869 

498 
694 
889 

5i8 
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908 

537 
733 
928 

557 
753 
947 

577 
772 
967 

596 
792 

986 

616 
811 

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224 
225 
226 

35  025 
218 
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238 
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257 

449 

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276 
468 

102 
295 

488 

122 
3.5 
507 

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334 
526 

160 
353 
545 

180 

372 

564 

199 

392 
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227 
228 
229 

230 

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660 

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698 

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774 

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36  173 

192 

211 

229 

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267 

286 

305 

324 

342 

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0 

1 

2 

3 

4 

5 

6 

7 

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0 

230-260 


1  N 

0 

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4 

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232 

233 

234 

235 
236 

237 
238 
239 

210 

241 
242 
243 

244 
245 

246 

247 
248 
249 

250 

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262 
253 

254 
255 
256 

257 
258 
259 

260 

36  173 

192 

21 1 

229 

248 

267 

286 

305 

324 

342 

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549 
736 

922 

37  107 
291 

475 
658 
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568 
754 

940 

125 

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493 
676 

858 

399 
586 
773 

959 

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328 

5ii 
694 
876 

4i8 
605 
791 

977 
162 
346 

53o 
712 

894 

436 
624 
810 

996 

181 
365 

548 
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455 

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199 

383 

566 

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585 
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493 
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785 
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039 

057 

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166 

184 

202 
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270 
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287 
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256 
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292 
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358 
533 
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489 
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199 

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507 
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393 

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346 

525 
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881 

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585 
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364 
543 
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252 

428 
602 

777 

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863 

881 

898 

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967 
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654 
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329 

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347 

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196 
363 

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192 
364 

535 
705 

875 

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212 
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209 
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552 
722 
892 

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229 
397 

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226 
398 

569 
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246 
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415 

586 
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432 

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447 

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278 
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620 
790 
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296 
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295 
466 

637 
807 
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497 

5i4 

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547 

564 

58i 

597 

6i4 

63i 

647 

N 

0 

1 

2 

S 

4 

5 

G 

7 

8 

9 

PP 

I 
2 
3 

4 
5 
6 

7 
8 

9 

19 

1.9 

3.8 

5.7 

7.6 

9.5 

H.4 

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l5.2 

17. 1 

18 

1.8 
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12.6 
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1-7 
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10.2 

1 1.9 
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2 
3 

4 

5 
6 

7 
8 
9 

16 
1.6 

3.2 

4.8 

6.4 
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1 1.2 

12.8 
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15 

1.5 
3.0 
4.5 

6.0 
7-5 
9.0 

10.5 
12.0 
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14 

1.4 

2.8 
4.2 

5.6 
7.0 
8.4 

9.8 
1 1.2 

12.6 

260-300 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

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261 
262 
263 

4i  497 

5i4 

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547 

564 

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597 

614 

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647 

664 
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681 
847 

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863 

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714 

880 

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896 

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764 

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780 

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797 
963 

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364 
265 
266 

42  160 
325 

488 

177 
341 

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193 

357 

521 

210 

374 
537 

226 
390 

553 

243 
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570 

269 

423 

586 

275 
439 
602 

292 
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267 
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269 

270 

271 
272 
273 

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846 

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700 
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716 

878 
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782 

894 

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765 

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797 
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169 

185 

201 

217 

233 

249 

265 

281 

297 
457 
616 

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473 
632 

329 

489 
648 

345 
505 

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521 

680 

377 
537 
696 

393 
553 
712 

409 
569 
727 

425 
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274 
275 
276 

775 

933 

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949 
107 

807 

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122 

823 
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170 

870 

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886 

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201 

902 

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217 

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232 

277 
278 
279 

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281 
282 
283 

248 

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264 
420 
576 

279 
436 
592 

295 

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607 

3ii 
467 
623 

326 

483 
638 

342 

498 
654 

358 
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373 
529 
685 

389 

545 
700 

716 

731 

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762 

778 

793 

809 

824 

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855 

87. 

45  025 

179 

886 
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194 

902 
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209 

917 
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225 

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086 
240 

948 
102 
255 

963 
117 
271 

979 
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286 

994 
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3i7 

284 
285 
286 

332 
484 
637 

347 
500 
652 

362 

515 
667 

378 
53o 
682 

393 

545 
697 

4o8 
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712 

423 

576 
728 

439 
591 
743 

454 
606 
758 

469 
621 

773 

287 
288 
289 

290 

291 
292 
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788 

939 

46  090 

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954 
105 

818 
969 
120 

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135 

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150 

864 

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165 

879 

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180 

894 

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909 

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210 

924 

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225 

24o 

255 

270 

285 

3oo 

315 

33o 

345 

359 

374 

389 
538 
687 

4o4 
553 
702 

419 
568 
716 

434 
583 
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449 
598 
746 

464 
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761 

479 
627 

776 

494 

642 

790 

509 
657 
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523 
672 
820 

294 
295 
296 

835 

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47  129 

850 

997 
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864 

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159 

879 

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173 

894 

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188 

909 

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202 

923 

♦070 

217 

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232 

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246 

967 

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261 

297 
298 
299 

300 

276 
422 
567 

290 

436 
582 

805 

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596 

319 

465 
611 

334 

480 
625 

349 

494 
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363 
509 

654 

378 

524 
669 

392 

538 
683 

407 
553 
698 

712 

727 

74 1 

756 

770 

784 

799 

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828 

842 

N 

0 

1 

2. 

3 

4 

6 

6 

7 

8 

0 

300—330 


N 

800 

3oi 
3o2 
3o3 

3o4 
3o5 
3o6 

3o7 
3o8 
309 

310 

3ii 

3l2 

3i3 

3i4 
3i5 
3i6 

3i7 
3i8 
319 

320 

321 
322 

323 

324 

325 

326 

327 

328 

329 
330 


O 

4?  712 

857 

48  001 

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287 
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572 

714 
855 
996 


49  1 36 


727 

871 
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728 
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173 

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601 

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756 

900 

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615 

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178 


770 

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344 
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911 

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192 


5 

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216 

359 
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643 

785 

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206 


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799 

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2.3  o 

373 

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8i3 

958 
101 
244 

387 
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671 

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954 
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8   9 

828 

972 
116 
259 

4oi 
544 
686 

827 

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234 


248 


276 
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554 

693 
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969 

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243 
379 


290 

429 
568 

707 

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120 
256 
393 


3o4 
443 
582 

721 
859 
996 

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270 
406 


3i8 
457 
596 

734 
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305 
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296 

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547 
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771 
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636 
723 
810 

644 
732 
819 

653 

740 
827 

662 

749 
836 

671 

758 
845 

679 
767 
854 

688 
775 
862 

697 
784 
871 

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793 

880 

714 
801 

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928 

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949 

958 

966 

975 

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502 

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70  070 
157 

079 
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088 
174 

096 

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105 
191 

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200 

122 
209 

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217 

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226 

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234 

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243 
329 
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252 

338 

424 

260 
346 

432 

269 

355 
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278 
364 
449 

286 
372 
458 

295 
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467 

3o3 
389 
475 

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398 
484 

321 

4o6 
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510 

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586 
672 

509 

595 
680 

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689 

526 
612 
697 

535 
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544 
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714 

552 
638 
723 

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646 
731 

569 
655 

740 

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663 

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774 

783 

791 

800 

808 

817 

825 

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0 

1 

2 

8 

4 

5 

6 

7 

8 

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510-540 


N 

0 

1 

2 

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4 

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927 

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935 
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859 
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876 
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885 
969 
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978 
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902 
986 
071 

910 

995 
079 

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096 
181 
265 

105 
189 
273 

ii3 

198 
282 

122 
206 
290 

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299 

139 

223 

307 

147 

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240 
324 

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248 
332 

172 

257 
341 

5i7 
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520 

521 

522 

523 

349 
433 
5i7 

357 

44i 
525 

366 
450 
533 

374 
458 
542 

383 
466 
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391 

475 
559 

399 
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600 

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659 

667 

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684 
767 
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692 

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700 

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709 

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875 

717 
800 
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725 
809 
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734 
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742 
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834 
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123 

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999 
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532 
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181 
263 
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189 
272 

354 

198 
280 
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288 

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296 

378 

222 
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387 

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395 

239 

321 

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247 
329 
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255 
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509 
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542 
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632 
713 

558 
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722 

567 
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575 
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535 
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770 
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255 

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272 

280 

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296 

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0.9 
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7 
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540—580 


N 

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543 

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376 
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600 

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547 
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320 

170 
249 
327 

178 
257 

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265 
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570 

421 
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557 
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562 
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601 
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796 

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850 

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881 

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912 
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097 

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105 

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564 
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143 
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228 
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236 

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166 
243 
320 

174 
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328 

182 
259 
335 

189 
266 
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197 
274 
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567 
568 
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570 

571 
572 
573 

358 
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366 
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519 

374 
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526 

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389 
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397 
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686 
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694 
770 
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702 

778 

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709 
785 
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717 
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732 
808 
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574 
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110 

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580 

118 
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200 
275 

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208 
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290 

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170 

245 
320 

178 
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260 
335 

343 

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358 

365 

373 

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388 

395 

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N 

0 

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3 

4 

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6 

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580—610 


N 

0 

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2 

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4 

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358 

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545 
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485 
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166 
240 
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820 

181 

254 
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188 
262 
885 

195 
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342 

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349 

210 

288 
357 

217 
291 
364 

225 
298 
371 

594 
595 
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379 
452 
525 

886 
459 
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898 
466 
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422 

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670 

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605 
677 
750 

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634 
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714 
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924 
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089 

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118 
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211 

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219 
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161 
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168 

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607 
608 
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888 

405 
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340 

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419 
490 

355 
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362 
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376 

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519 

383 

455 
526 

533 

540 

547 

554 

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N 

0 

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3 

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4 

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6 

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0.8 
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4 
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7 
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4.9 
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4.2 

4.8 
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610 

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3 

4 

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6 

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540 

547 

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682 
753 

618 
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760 

625 
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633 

704 
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711 

781 

647 
718 
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725 
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661 
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668 
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6i4 
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817 

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902 
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909 

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930 

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617 
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120 
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211 

078 
1 48 
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225 

092 
162 

232 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

309 

379 
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3i6 

386 
456 

323 
393 

463 

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470 

337 
407 
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344 

4i4 

484 

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421 

491 

358 
428 
498 

365 
435 
505 

372 

442 

5ii 

624 
625 
626 

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588 
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525 

595 
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602 
671 

539 

609 

678 

546 
616 
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553 
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699 

567 
637 
706 

574 
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720 

627 
628 
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630 

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632 
633 

727 
796 
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734 
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74i 
810 
879 

748 
817 
886 

754 
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761 
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900 

768 
837 
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775 
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920 

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969 

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996 

80  oo3 
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010 
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024 
092 
161 

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168 

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106 
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182 

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120 

188 

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127 
195 

065 
1 34 
202 

634 
635 
636 

209 
277 
346 

216 

284 
353 

223 
291 
359 

229 

298 
366 

236 

305 
373 

243 

3l2 

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387 

257 
325 
393 

264 
332 
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271 
339 
407 

637 
638 
639 

640 

64 1 
642 
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421 
489 
557 

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496 

564 

434 
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570 

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509 
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448 
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455 
523 
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462 
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598 

468 
536 
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475 
543 
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618 

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645 

652 

659 

665 

672 

679 

686 
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693 
760 

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767 
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7.3 

781 
848 

720 
787 
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726 

794 
862 

733 
801 
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740 
808 
875 

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644 
645 
646 

889 
956 

81  023 

895 
963 
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902 
969 
037 

909 

976 
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916 
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922 
990 
057 

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070 

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084 

647 
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097 
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171 
238 

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178 
245 

117 

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124 
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258 

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198 
265 

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271 

1 44 
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278 

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285 

291 

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325 

33i 

338 

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N 

0 

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N 

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652 
653 

654 
655 
656 

657 
658 
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662 
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666 

667 
668 
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670 

671 
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673 

674 
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677 
678 
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680 

81  291 

298 

305 

3ii 

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33i 

338 

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35i 

358 
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558 
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757 
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365 

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498 

564 
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697 

763 
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371 
438 
505 

571 
637 
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770 
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378 

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710 

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385 
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657 
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598 
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617 

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82  020 
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217 
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289 
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419 
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536 
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620 

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646 

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659 

666 

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737 
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866 

930 

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123 

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692 
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174 
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860 

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270 

276 

283 

289 

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2.8 
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4.9 
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2 

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6 

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3.0 

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4.8 
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18 


680 

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0 

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1 

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759 

763 

768 

772 

777 

782 

786 

791 

795 

800 

8o4 

809 

8i3 

818 

823 

827 

832 

836 

84 1 

845 

85o 

855 

859 

962 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

954 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

955 

98  000 

005 

009 

oi4 

019 

023 

028 

o32 

037 

o4i 

966 

o46 

o5o 

055 

059 

o64 

068 

073 

078 

082 

087 

957 

091 

096 

100 

105 

109 

ii4 

118 

123 

127 

I  32 

958 

137 

i4i 

i46 

i5o 

«55 

159 

1 64 

168 

173 

177 

969 
960 

182 

186 

191 

.95 

200 

204 

209 

2l4 

218 

223 

227 

232 

236 

24 1 

245 

250 

254 

259 

263 

268 

N 

0 

1  1  2 

3 

4 

5 

0 

7 

8 

9 

PP      5 

t 

i 

I 

0.5 

I 

0 

4 

2 

1.0 

2 

0 

8 

3 

1.5 

3 

I 

2 

4 

2.0 

4 

I 

6 

5 

2.5 

5 

2 

0 

6 

3.0 

6 

2 

4 

7 

3.5 

7 

2 

8 

8 

4.0 

8 

3 

2 

^__ 

9 

4.5 

9 

3 

6 

26 


960—1000 


N 

0 

1 

2 

3 

4 

6 

6 

7 

8   9  1 

OGO 

961 
962 
963 

98  227 

232 

236 

24l 

245 

250 

254 

259 

263 

268 

272 
3i8 
363 

277 
322 
367 

281 
327 
372 

286 
33i 
376 

290 
336 
38i 

295 
340 

385 

299 

345 
390 

3o4 
349 
394 

3o8 
354 
399 

3i3 
358 
4o3 

964 
965 

9G6 

4o8 
453 
498 

4l2 

457 

502 

417 
462 
507 

421 
466 
5ii 

426 

471 
5i6 

43o 

475 

520 

435 
48o 
525 

439 

484 
529 

444 
489 
534 

448- 
493 
538 

967 
96S 
969 

970 

971 
972 
973 

543 
588 
632 

547 
592 

637 

552 
597 
64i 

556 
601 

646 

56i 
6o5 
65o 

565 
610 
655 

570 

6i4 
659 

574 
619 
664 

579 
623 

668 

583 
628 
673 

677 

682 

686 

691 

695 

700 

704 

709 

7i3 

717 

722 
767 
811 

726 
771 
816 

73i 
776 
820 

735 
780 
825 

740 
784 
829 

744 
789 
834 

749 
793 
838 

753 
798 
843 

758 
802 
847 

762 
807 

85i 

974 
976 

856 
900 
945 

860 

905 
949 

865 
909 

954 

869 
914 
958 

874 
918 
963 

878 
923 
967 

883 
927 
972 

887 
932 
976 

892 
936 
981 

896 
941 
985 

977 
978 

979 

980 

981 
982 

989 

99  o34 

078 

994 
o38 
o83 

998 
043 
087 

*oo3 
047 
092 

*oo7 
o52 
096 

*OI2 

o56 
100 

*oi6 
061 
105 

*021 
065 
109 

*025 

069 
ii4 

*029 

074 
118 

123 

127 

i3i 

i36 

i4o 

145 

149 

i54 

i58 

162 

167 

21  I 

171 
216 

176 
220 

180 
224 

185 
229 

189 
233 

193 

238 

198 
242 

202 

247 

207 

25l 

983 

255 

260 

264 

269 

273 

277 

282 

286 

291 

296 

984 
985 
986 

3oo 

344 

388 

3o4 
348 
392 

3o8 
352 
396 

3i3 
357 
4oi 

3i7 
36i 
4o5 

322 

366 
4io 

326 
370 

4i4 

33o 

374 
419 

335 
379 

423 

339 

383 
427 

987 
988 
989 
990 

991 
992 
993 

432 

476 
520 

436 

48o 
524 

44i 
484 
528 

445 
489 
533 

449 
493 
537 

454 
498 
542 

458 
5o2 
546 

463 
5o6 
55o 

467 

5ii 
555 

471 
5i5 
559 

564 

568 

572 

577 

58i 

585 

590 

594 

5^9 

6o3 

607 
65i 
695 

612 
656 
699 

616 
660 
704 

621 
664 
708 

625 
669 
712 

629 
673 
717 

634 
677 
721 

638 
682 
726 

642 

686 
730 

647 
691 
734 

994 
995 
996 

739 

782 
826 

743 
787 
83o 

747 
791 
835 

752 
795 
839 

756 
800 

843 

760 
8o4 
848 

765 
808 
852 

769 
8i3 
856 

774 
8.7 
861 

778 
822 
865 

997 
998 
999 

1000 

N 

870 
913 
957 

874 
917 
961 

878 
922 
965 

883 
926 
970 

887 
930 
974 

891 

935 

978 

896 
939 

983 

900 

944 
987 

904 
948 
991 

909 
952 
996 

00  000 

oo4 

009 

oi3 

017 

022 

026 

o3o 

035 

039 

0 

1 

2 

3 

4 

5 

6 

7 

8   9  J 

27 


TABLE  II 

FIVE -PLACE     LOGARITHMS 

OF    THE 

TRIGONOMETRIC     FUNCTIONS 


TO  EVERY  MINUTE 


0°. 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

60 

59 

0 

— 

30103 
17609 

— 

— 

0 .  00  000 

1 

6.46  373 

6.46373 

3.53627 

0.00  000 

2 

6.76  476 

6.76  476 

30103 

3.23  524 

0.00  000 

58 

3 

6.94  085 

6.94085 

17609 

3.o5  915 

0.00  000 

57 

12494 

13494 

4 

7.06  579 

9691 

7.06  579 

9691 

2.93  421 

0.00  000 

56 

b 

7. 16  270 

7918 

7. 16  270 

2.83  730 

0.00  000 

55 

6 

7.24  188 

7.24  188 

7918 

2.75  812 

0.00  000 

54 

6694 

6694 

7 
8 

7.30882 
7.36682 

5800 

7.30882 
7.36682 

5800 

2.69  118 
2.633i8 

0.00  000 
0.00  000 

53 

52 

9 

7.41  797 

5"5 
4576 

7.41  797 

5"5 
4576 

2.58  2o3 

0.00  000 

5i 

10 

7.46373 

4139 
3779 
3476 

7.46373 

4139 
3779 

2.53  627 

0.00  000 

50 

1 1 

12 

7.5o  5i2 
7.54  291 

7.5o  5i2 
7.54  291 

2.49488 
2.45  709 

0.00  000 
0.00  000 

49 

48 

i3 

7.57  767 

7.57  767 

3476 

2.42  233 

0.00  000 

47 

i4 
i5 
i6 

7.60985 
7.63  982 
7.66  784 

3218 
2997 
2802 
2633 

7.60  986 
7.63  982 
7.66785 

3219 
2996 
2803 
2633 

2.39  oi4 
2.36  018 
2.33  215 

0.00  000 
0.00  000 
0.00  000 

46 

45 
44 

I? 

7.69  417 

2483 

7.69418 

2482 

2.3o582 

9.99999 

43 

i8 

7.71  900 

2348 

7.71  900 

2348 
2228 

2119 

2.28  100 

9.99999 

42 

'9 

7.74  248 

7.74  248 

2.25   752 

9.99999 

4i 

20 

7.76475 

2227 
2119 

7.76  476 

2.23  524 

9.99999 

40 

21 

7.78  594 

7.78  595 

2.2  1   405 

9.99999 

39 

22 

7.80  615 

7.80615 

2.19385 

9.99999 

38 

23 

7.82545 

1848 
1773 

7.82  546 

«93i 
1848 

1773 

2.  17  454 

9.99999 

37 

24 

7.84393 

7.84394 

2.  I  5  606 

9.99999 

36 

2!) 

7.86  166 

7.86  167 

2.i3  833 

9.99999 

35 

26 

7.87870 

1704 
1639 

7.87  871 

1639 

2.12  129 

9.99999 

34 

2  7 

7.89  509 

7.89  5io 

1579 

2 . 1  o"  490 

9.99999 

33 

28 

7.91  088 

7.91  089 

2,08  911 

9.99999 

32 

29 

7.92  612 

7.92  6i3 

2.07  387 

9.99998 

3i 

1473 

30 

7.94  084 

7.94  086 

2.o5  914 

9.99998 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin. 

/  - 

89° 

30.                                                  1 

PP 

.1 

9691 

4576 

9997 

.1 

9483 

3119 

1848 

1704 

1579 

1472 

969 

458 

300 

248 

212 

.85 

.1 

170 

158 

'47 

.2 

'938 

91S 

599 

.2 

497 

424 

370 

.3 

34" 

316 

294 

•3 

2907 

1372 

899 

•3 

745 

636 

554 

•3 

5" 

474 

442 

•4 

3876 

1830 

1 199 

•4 

993 

848 

739 

•4 

682 

632 

589 

•  5 

4846 

2288 

1498 

•5 

1242 

1060 

924 

•5 

852 

789 

736 

.6 

581S 

2646 

1798 

.6 

1490 

1271 

1 109 

.6 

I022 

947 

883 

.7 

6784 

3203 

2098 

•7 

1738 

1483 

1294 

•7 

"93 

iios 

1030 

.8 

7753 

3661 

2398 

.8 

1986 

1695 

1478 

.8 

«3<>3 

1263 

1 178 

8722    41 18  i 

^90^ 

1663 

3o 


0°  30 . 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

30 

7.94  o»4 

1424 

7.94  086 

1424 

2.o5  914 

9.99998 

30 

■6i 

7.95  5o8 

1379 
1336 

7.95  5io 

'379 

2.o4  490 

9- 

99998 

29 

32 

7.96887 

7.96  889 

2.o3   III 

9- 

99998 

28 

Si 

7.98  223 

7.98  225 

2.01  775 

9- 

99998 

27 

1297 

1297 

34 

7.99  520 

1259 

7.99  522 

1 259 

2.00  478 

9- 

99998 

26 

3b 

8.00  779 

8.00  781 

1.99219 

9- 

99998 

25 

36 

8.02  002 

1190 

8 .  02  oo4 

1190 

1.97996 

9- 

99998 

24 

3? 

8.o3  192 

1158 

8.o3  194 

1159 

1.96806 

9- 

99997 

23 

38 

8.o4  35o 

1128 

8,04353 

1128 

1 .95  647 

9- 

99  997 

22 

39 

8.05478 

1100 

8.o5  48i 

IIOO 

1.94  5 19 

9- 

99  997 

21 

40 

8.06  578 

8.06  58i 

1 .93  419 

9- 

99  997 

20 

4i 

8.07  650 

1046 

8.07  653 

1047 

1 .92  347 

9- 

99  997 

•9 

42 

8.08  696 

8.08  700 

1 .91  3oo 

9- 

99997 

18 

43 

8.09  718 

999 

8.09  722 

998 

1 .90  278 

9- 

99  997 

£7 

44 

8.10  717 

8. 10  720 

976 

1.89  280 

9- 

99996 

16 

45 

8. 1 1  693 

8.  II  696 

1.88  3o4 

9- 

99996 

i5 

46 

8.12  647 

8.12  65i 

9SS 

1.87  349 

9- 

99996 

i4 

47 

.8.i358i 

8.13585 

934 

1. 86415 

9- 

99996 

i3 

48 

8.14495 

896 
877 

8. 1 4  500 

89s 

878 

860 
843 

1.85  5oo 

9- 

99996 

12 

49 

8.i5  391 

8.1 5  395 

1.84  605 

9- 

99996 

II 

50 

8.16268 

8.16  273 

I .83  727 

9- 

99995 

10 

5i 

8.17  128 

Sat. 

8.17  i33 

1.82  867 

9- 

99995 

9 

52 

8.17971 

827 
812 

8.17  976 

828 

1 .82  024 

9- 

99995 

8 

53 

8.18  798 

8.18804 

812 

1. 81  196 

9- 

99  995 

7 

54 

8. 19  610 

8.19616 

797 
782 

769 

756 

1.80  384 

9- 

99  995 

6 

55 

8.20  407 

8.2o4i3 

1.79587 

9- 

99994 

5 

56 

8.21  189 

782 
769 

8.21  195 

1.78805 

9- 

99994 

4 

5? 

8.21  958 

8.21  964 

1,78036 

9- 

99994 

3 

58 

8.22  713 

8.22  720 

1 .77  280 

9- 

99994 

2 

59 

8.23  456 

743 
73° 

8.23  462 

742 
730 

1.76  538 

9 

99994 

I 

60 

8.24  i86 

8.24  192 

1.75808 

9 

99993 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

' 

8J 

r. 

PP 

1379 

1323 

IIOO 

.1 

999 

9M 

860 

.1 

813 

769 

730 

.1 

n8 

122 

no 

100 

91 

86 

81 

77 

73 

.2 

276 

245 

220 

.2 

200 

■83 

172 

.2 

162 

154 

146 

•3 

414 

367 

330 

•3 

300 

274 

258 

•3 

244 

231 

219 

•4 

552 

48q 

440 

•4 

400 

366 

344 

•4 

32s 

308 

292 

•5 

690 

612 

550 

•5 

500 

457 

430 

•5 

40t 

385 

365 

.6 

827 

734 

660 

.6 

599 

548 

S16 

.6 

487 

461 

438 

•7 

06s 

856 

770 

7 

699 

640 

602 

•7 

S6J 

538 

5" 

.8 

1 103 

978 

880 

.8 

799 

73' 

688 

.8 

6sc 

615 

S84 

-ii;2. 

1241    '    IIOI 

.J22- 

_2^ 

692 

.6^2. 

3i 


10. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d.    1  L.  Cotg. 

L.  Cos. 

0 

8.24  186 

717 

8.24  192 

7.8 
706 

1.75808 

9.99993 

60 

I 

8.24903 

8.24  910 

I  .75  090 

9- 

99  993 

59 

2 

8.25  609 

695 
684 
67:1 

8.25  616 

1.74  384 

9- 

99993 

58 

3 

8.26  3o4 

8.26  3l2 

696 

68.; 
673 

1.73688 

9- 

99993 

57 

4 

8.26988 

8.26  996 

1 .73  oo4 

9- 

99992 

56 

5 

8.27661 

8.27  669 

1.72  33i 

9- 

99  992 

55 

6 

8.28  324 

663 

6S3 
644 

8.28  332 

663 
654 
643 

1. 71  668 

9- 

99992 

54 

7 

8.28  977 

8.28986 

1 .71  oi4 

9- 

99992 

53 

8 

8.29  621 

634 

8.29  629 

I .70  371 

9- 

99992 

52 

9 

8.30255 

8.30  263 

634 

1.69737 

9- 

99991 

5i 

10 

8.30879 

624 
616 

8.3o888 

617 
607 

I .69  112 

9- 

99991 

50 

1 1 

8.3i  495 

608 

8.3i  505 

1.68495 

9- 

99991 

49 

12 

8.32  io3 

8.32  112 

1.67888 

9- 

99990 

48 

i3 

8.32  702 

599 

8.32  711 

599 

1 .67  289 

9- 

99990 

47 

i4 

8.33  292 

583 

8.33  3o2 

<:84 

1.66698 

9- 

99990 

46 

i5 

8.33875 

8.33  886 

1.66  ii4 

9- 

99990 

45 

i6 

8.34450 

568 

8.34461 

575 
568 

561 

1.65  539 

9- 

99989 

44 

'7 

8.35  018 

560 

8.35  029 

1.64  971 

9- 

99989 

43 

i8 

8.35578 

8.35  590 

1 .64  4io 

9- 

99989 

42 

•9 

8.36  i3i 

8.36  143 

553 

1.63  857 

9- 

99  989 

4i 

20 

8.36678 

8.36689 

546 

1.63  3ii 

9- 

99  988 

40 

21 

8.37  217 

533 

8.37  229 

533 

1 .62  771 

9- 

99988 

39 

22 

8.37750 

526 

8.37  762 

1.62  238 

9- 

99  988 

38 

23 

8.38  276 

8.38289 

527 

1 .61  711 

9- 

99987 

37 

520 

520 

24 

8.38  796 

514 

8.38809 

514 

1 .61  191 

9- 

99987 

36 

25 

8.39310 

508 

502 
496 

8.39  323 

1 .60  677 

9- 

99987 

35 

26 

8.39818 

8.39832 

509 

1.60  i68 

9- 

99  986 

34 

27 

8.4o  320 

8.4o334 

496 

1.59666 

9 

99  986 

33 

28 

8.40816 

8.40  83o 

1.59  170 

9- 

99  986 

32 

29 

8.4i  307 

49' 

485 

8.4i  321 

491 

486 

1.58679 

9 

99985 

3i 

30 

8.4i  792 

8.4i  807 

1.58  193 

9 

99  985 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

' 

88° 

30.                                                  1 

PP 

.1 

706 

663 

634 

.1 

599 

575 

553 

533 

514 

496 

70.6 

66.3 

^n 

59-9 

57-5 

55-3 

.1 

53-3 

51-4 

49.6 

•  .2 

141. 2 

132.6 

126.8 

.2 

1 19. 8 

115-0 

110.6 

.2 

io6.t 

102.8 

99.2 

•3 

211.8 

198.9 

190.2 

•3 

179.7 

172-5 

165.9 

-3 

159-9 

•54-2 

48.8 

•4 

282.4 

265.2 

253.6 

•4 

239.6 

230.0 

221.2 

•4 

213.5 

205.6 

98.4 

■5 

3530 

33'S 

317.0 

•5 

2995 

287.5 

276.5 

•5 

266.= 

257.0    : 

48.0 

.6 

423.6 

397-8 

380.4 

,6 

359-4 

3450 

33'-8 

.6 

3'9-t 

308.4     = 

97.6 

•7 

494.2 

4641 

443-8 

•7 

4>9-3 

402.5 

387- > 

•7 

373-' 

359-8    : 

47.2 

.8 

.S64.« 

530-4 

507.2 

.8 

479.2 

460.0 

442.4 

.8 

426.4 

4112    : 

96.8 

522.^ 

.9      479-7  '  462  6  1  .( 

46.4 

32 


1°30. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg.       L.  Cos. 

30 

8.4i  792 

8.4i  807 

1 .5s  193      9-99  9^5 

30 

3i 

8.42  272 

8.42  287 

1.57  713 

9- 

99985 

29 

32 

8.42  746 

8.42  762 

1.57238 

9- 

99  984 

28 

33 

8.43216 

470 
464 

8.43  232 

470 
464 

1.56768 

9- 

99  984 

27 

34 

8.43680 

8.43696 

460 

1.56  3o4 

9- 

99  984 

26 

35 

8.44  139 

459 

8.44  i56 

1.55  844 

9- 

99  983 

25 

36 

8.44594 

455 

8.44  611 

455 

1.55389 

9- 

99983 

24 

37 

8.45  o44 

8.45061 

446 

1.54  939 

9- 

99  983 

23 

38 

8.45489 

445 

8.45507 

1.54493 

9- 

99  982 

22 

^9 

8.45  930 

441 
436 

433 

8.45  948 

441 
437 

432 
428 

I .54  o52 

9- 

99982 

21 

40 

8.46  366 

8.46  385 

i.536i5 

9- 

99  982 

20 

4i 

8.46  799 

8.46817 

1.53  i83 

9- 

99981 

19 

42 

8.47  226 

8.47  245 

1.52  755 

9- 

99981 

18 

43 

8.47  650 

424 

8.47  669 

424 
420 

416 

1.52  33i 

9- 

99981 

'7 

44 

8.48069 

Alf, 

8.48089 

1 .5i  911 

9- 

99  980 

16 

45 

8.48485 

8.48  5o5 

i.5i  495 

9- 

99  980 

i5 

46 

8.48896 

411 

408 

8.48917 

412 
408 

i.5i  o83 

9- 

99  979 

i4 

47 

8.49  3o4 

8.49325 

I   50675 

9- 

99979 

i3 

48 

8.49708 

8.49  729 

I .5o  271 

9- 

99  979 

12 

49 

8.5o  108 

400 

8.5o  i3o 

401 

I .49  870 

9- 

99978 

1 1 

396 

397 

50 

8.5o  5o4 

393 

8.5o527 

393 

1-49  473 

9- 

99978 

10 

5i 

8.50897 

390 
386 
382 

8.5o  920 

1 .49  080 

9- 

99  977 

9 

52 

8.5i  287 

8.5i  3io 

386 
383 

1.48690 

9- 

99  977 

8 

53 

8.5i  673 

8.5i  696 

1 .48  3o4 

9- 

99  977 

7 

54 

8.52  o55 

8.52  079 

380 
376 

1.4?  921 

9- 

99976 

6 

55 

8.52434 

8.52459 

1.47  54i 

9 

99976 

5 

56 

8.52  810 

376 

8.52  835 

1.47  i65 

9 

99975 

4 

373 

373 

^7 

8.53  i83 

369 

8.53208 

370 

1 .46  792 

9 

99  975 

3 

58 

8.53  552 

8.53578 

1 .46  422 

9 

99974 

2 

59 

8.53  919 

367 
363 

8.53945 

367 
363 

1.46  o55 

9 

99974 

I 

60 

8.54  282 

8.54  3o8 

I .45  692 

9 

99  974 

0 

L.  Cos. 

d. 

L.  Cotg. 

d.      L.  Tang. 

L.Sin.    1        1 

8J 

s°.                                      1 

PP 

.1 

470     455 

441 

.1 

424 

408 

396 

.1 

386 

376 

367 

47.0 

45-5 

44.1 

42.4 

40.8 

39-6 

38- 

5       37-6 

36-7 

.2 

94.0 

91  0 

88.2 

.2 

84.8 

81.6 

xjr.8 

.2 

77- 

2         75.2 

73.4. 

■3 

141. 0 

136-5 

132.3 

•3 

127.2 

122.4 

•3 

"5 

i     112.8 

10. 1 

•4 

188.0 

182.0 

176.4 

-4 

169.6 

163.2 

158.4 

•4 

154 

♦     150.4 

46.8 

•5 

235-0 

227-5 

220.5 

•5 

212.0 

204.0 

198.0 

•5 

193- 

D       188.0 

<83-5 

6 

282.0 

273.0 

264.6 

.6 

254.4 

244.  b 

237-6 

.6 

231. 

5    225.6 

220.2 

•7 

329.0 

^i8.s 

308.7 

•7 

296.8 

285.6 

277.2 

•7 

270. 

2       263.2 

256.9 

.8 

376.0 

364.0 

352.8 

.8 

339-2 

3264 

316.8 

.8 

308. 

8    300.8 

293.6 

•  9      4230 

409.5 

.9  '  381.6  ; 

367.2 1 356.4 

33 


10 


20 


22 
23 

24 
25 
26 

27 

28 

29 
30 


L.  Sin. 

8.54  282 

8.54  642 

8.54  999 

8.55  354 

8.55  705 

8.56  o54 

8.56  4oo 

8.56743 

8.57  o84 
8.57  421 


8.57  757 


,58  089 
,58  419 
.58747 

,59  072 
,59  395 
.59  715 

,60  o33 
,60  349 
.60  662 


8.60  973 


1.61  282 

1.61  589 
i.6i  894 

1.62  196 
i.62  497 
i.62  795 

i.63  091 
i.63  385 

1.63  678 


8.63968 


d. 

360 
357 
355 

35' 
349 
346 
343 
341 
337 
336 
332 
330 
328 
325 
323 
320 

318 
316 
313 
3" 
309 
307 
305 
302 
301 
298 
296 
294 

293 
290 


L.  Tang. 

8.54308 


8.54669 

8.55  027 
8.55  382 

8.55  734 

8.56  o83 
8.56  429 

8.56773 
8.57  ii4 
8.57452 

8.57788 

8.58  121 
8.58  451 
8.58  779 

8.59  io5 

8.59428 
8.59  749 

8.60068 
8.60  384 
8.60698 

8.61  009 

8.61  319 
8.61  626 
8.61  931 

8.62  234 

8.62535 
8.62834 

8.63  i3i 
8.63426 
8.63718 

8.64  009 


361 
358 

355 
352 
349 
346 
344 
341 
338 
336 

333 
330 
328 
326 
323 
321 
3'9 
316 
314 
3" 
310 

307 
30s 
303 
301 
299 
297 
29s 
292 
291 


L.  Cotg. 

I .45  692 

.45  33i 
.44  973 
.44618 

.44266 
•  43  917 
.43571 

.43  227 
.42  886 
.42  548 


.42  212 


,4i  879 
,4i  549 
.4i  221 

,40895 
,40  572 

,40  25l 

.39  932 
.39  616 
.39  3o2 


,3.8  991 


,38  681 
,38  374 
,38069 

,37  766 
,37465 
,37  166 

,36869 
,36  574 
.36  282 


.35  991 


L.  Cos. 

9.99974 


99973 
99973 
99972 

99972 
99971 
99971 

99970 
99970 
99969 


99969 


99  968 
99  968 
99967 

99967 
99967 
99  966 

99  966 
99965 
99  964 


99  964 


99  963 
99  963 
99  962 

99  962 
99961 
99961 

99  960 
99  960 
99959 


99959 


L.  Cos. 


L.  Cotg.  1    d. 


L.  Tang. 


L.  Sin. 


87°  30'. 


pp 

360 

350 

340 

.1 
.2 

•3 

330 

330 

310 

.1 
.2 
•3 

300 

390 

385 

.  I 

.2 

•3 

36 
72 
108 

35 
70 
105 

I02 

99 

64 
96 

11 
93 

30 
60 
90 

5! 
87 

28.5 
57.0 
85.5 

•4 
•5 
.6 

144 
180 
216 

140 
175 

2IO 

136 
170 
204 

•4 

132 

165 
198 

128 
160 
192 

124 
155 
186 

•4 
•5 

.6 

120 

116 
145 
»74 

114.0 
142.5 
171.0 

•7 

.8 

352 

288 
324 

245 
280 
315 

938 
272 

306    __ 

•9 

231 

264 
297 

224 
256 
288 

279 

•7 
.8 

•9 

210 

240 
270  _ 

203 
232 
161 

199.5 
228.0 
2561S 

34 


2°  . 

30'. 

/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

30 

8.63  968 

8.64  009 

I .35  991 

9.99959 

30 

3i 

8.64  256 

287 

8.64298 

287 

1 .35  702 

9- 

99  958 

29 

32 

8.64  543 

8.64  585 

1.35  415 

9- 

99  958 

28 

33 

8.64827 

284 
283 

8.64  870 

285 
284 

1.35  i3o 

9- 

99  y57 

27 

34 

8.65  no 

281 

8.65  i54 

1.34  846 

9- 

99  956 

26 

35 

8.65  391 

8.65  435 

1.34565 

9- 

99  956 

25 

36 

8.65  670 

279 

8.65715 

278 
276 

1.34  285 

9- 

99955 

24 

3? 

8.65  947 

276 

8.65993 

1 .34  007 

9- 

99955 

23 

38 

8.66223 

8.66269 

1.33  731 

9- 

99  954 

22 

39 

8.66497 

274 
272 

8.66  543 

274 
273 

1.33457 

9- 

99954 

21 

4U 

8.66  769 

270 
269 

8.66816 

271 
269 
268 
266 

1.33  184 

9- 

99953 

20 

4i 

8.67  039 

8.67  087 

I .32  913 

9- 

99952 

19 

42 

8.67  3o8 

8.67  356 

1.32  644 

9- 

99952 

18 

43 

8.67575 

266 

8.67  624 

1.32  376 

9- 

99951 

17 

44 

8.67841 

263 

8.67890 

264 

1 .32  1 10 

9- 

99951 

16 

45 

8.68  io4 

8.68  i54 

i.3i  846 

9- 

99950 

i5 

46 

8.68  367 

263 
260 

8.68417 

263 
261 

i.3i  583 

9- 

99949 

i4 

47 

8.68627 

8.68678 

60 

I.3l   322 

9- 

99949 

i3 

48 

8.68  886 

259 

8.68938 

i.3i  062 

9- 

99  948 

12 

49 

8.69  i44 

258 

8.69  196 

258 

i.3o8o4 

9- 

99  948 

1 1 

50 

8.69  4oo 

254 

8.69453 

257 
255 

1 .3o  547 

9- 

99947 

10 

5i 

8.69654 

253 

8.69708 

1 .  3o  292 

9- 

99  946 

9 

52 

8.69  907 

8.69  962 

i.3oo38 

9- 

99  946 

8 

53 

8.70  159 

252 
250 

8.70  2l4 

251 

1 .29  786 

9- 

99945 

7 

54 

8.70  409 

8.70  465 

249 
248 
246 

1.29535 

9- 

99  944 

6 

55 

8.70658 

8.70  714 

1 .29  286 

9- 

99  944 

5 

56 

8.70  905 

247 
246 

8.70  962 

1 .29  o38 

9- 

99943 

4 

^1 

8.71  i5i 

8.71  208 

I .28  792 

9- 

99  942 

3 

58 

8.71  395 

8.71  453 

1.28547 

9- 

99  942 

2 

59 

8.71  638 

243 

8.71  697 

244 

1.28  3o3 

9- 

99941 

I 

60 

8.71  880 

8.71  940 

I .28  060 

9 

99  940 

0 

L.  Cos. 

d. 

L.  Cotg.  1    d.    1  L.  Tang. 

L.  Sin. 

f 

87°. 

1 

PP 

.1 

280 

375 

270 

.1 

265 

26.5 

260 

26.0 

255 

.1 

250 

a45 

240 

28.0 

27.5 

27.0 

25-5 

25- 1 

>      24.5 

24.0 

.2 

56.0 

55- 0 

54- 0 

.2 

53-0 

52.0 

51.0 

.2 

50.  c 

)      49.0 

48.0 

■3 

84.0 

82.5 

81.0 

•3 

79-5 

78.0 

76-5 

•3 

75-t 

>      73-5 

72.0 

•4 

112.0 

1 10.0 

108.0 

•4 

106.0 

104.0 

102.0 

•4 

100.  c 

)      98.0 

96.0 

•5 

140.0 

137-5 

135.0 

•5 

132-5 

130.0 

127-5 

■5 

125.C 

>     122.5 

120.0 

.6 

168.0 

165.0 

162.0 

.6 

159.0 

156.0 

153-0 

.6 

i5o.< 

>     147.0 

144.0 

•7 

196.0 

192-5 

189.0 

■7 

185-5 

182.0 

178.S 

•7 

17.5-C 

>     171. 5 

168.0 

.8 

224.0  1 220.0 

216.0 

.8 

212.0 

208  0 

204.0 

.8 

200.  c 

>     196.0 

192.0 

-;2- 

252.0  !  247.5 

243.0 

.9  \  238.5  1  234.0 

225.0  1  220.5  '    216.0    1 

35 


L.  Sin. 


d. 


L.Tang.  ;  d.   L.  Cotg. 


L.  Cos. 


10 


20 


21 
22 
23 

24 
25 
26 

27 
28 
29 


30 


b.71  880 


8. 


,72  120 
,72  359 
,72  597 

,72  834 
,  73  069 
,73  3o3 

73535 
73  767 
73997 


8.74  226 


.74454 
.74680 
.74  906 

.75  i3o 
.75  353 
.75  575 

.75  795 
.76  oi5 
.76234 


8.76451 


.76  667 
.76883 
.77097 

.77  3io 

.77  522 

•77  733 

,77943 
.78  l52 
.78  36o 


8.78  568 


340 
239 
238 
237 
235 
234 
232 

232 
230 
229 

228 
226 
326 
224 
223 

222 
220 
220 
219 
217 

216 
216 
214 
213 
212 
211 
210 
209 
208 
208 


8.71    940 


8.72  181 
8.72  420 
8.72    559 

8.72  896 

8.73  l32 
8.73  366 

8.73  600 
8.73  832 
8.74063 


8.74  292 


8.74521 
8.74748 

8.74  974 

8.75  199 
8.75423 
8.75645 

8.75  867 
8.76087 

8.76  3o6 


8,76  525 


8.76  742 
8.76958 

8.77  173 

8.77887 

8.77  600 
8.77811 

8.78  022 

8.78  232 
8.78441 


8.78  649 


241 
a39 
839 

237 
236 
234 
234 
232 
231 
229 

229 
227 
226 
225 
224 
222 
222 
220 
219 
219 

217 
ai6 
215 
214 
213 
211 
211 
210 
209 
208 


I  .28  060 


.27  819 
.27  58o 
.27  34i 

,27  io4 
.26868 
.26  634 

.26  4oo 

.26  168 
.25  937 


.25  708 


.25  479 

,25  252 

.25  026 

.24  801 
,24  577 
,24355 

,24  i33 
.23  913 
,23  694 


,23475 


.23  258 
.23  042 

.22  827 

.22  6l3 
.22  4oO 
.22   189 

.21  978 
.21  768 
.21   559 


,21  35i 


99  y4o 


99  940 
99939 
99  938 

99  938 
99937 
99  936 

99  936 
99  935 
99  934 


99  934 


99  933 
99  932 
99  932 

99931 
99  930 
99929 

99929 

99  928 
99927 


99  926 


99  926 
99925 
99  924 

99  923' 
99  923 
99  922 

99921 
99  920 
99  920 


60 

59 
58 
57 

56 
55 
54 
53 

52 

5i 


50 


49 
48 

47 

46 
45 
44 

43 
42 
4i 


40 


39 

38 
37 

36 
35 

34 

33 

32 

3i 


9.99919 


30 


L.  Cos. 


d. 


L.  Cot?. 


d. 


L.  Tang. 


L.  Sin. 


86°  30 . 


PP    238 


23.8 
47.6 
71.4 

J 
95.2 

119.0 

142.8 

166.6 
190.4 


«34 


23.4 
46.8 
70.2 

93-6 
117.0 
140.4 

163.8 
187.2 
210.6 


22.9 
45.8 
68.7 

91.6 
"4  5 
'37-4 

160.3 
183.2 
206.1 


22-5 

45.  o 
67-5 

90.0 
112.5 
«35o 

1 57- 5 
180.0 


iio.o 
132.0 


216 


21.6 
43  2 
64.8 

86.4 
108.0 
129.6 


154.0  151.3 
176.0  172.8 
108.0 


21.2 

42.4 
63.6 


106.0 
127.2 

148.4 
169.6 


20.8 

41.6 

62.4 


145.6 

166.4 
190.8  '  187.2 


20.4 

40.8 
61.3 


83  2    81.6 

104.0  102.0 
124.8  122.4 


142.8 

163.2 

183.6 


36 


3°, 

30'. 

/ 

L.  Sin.       d. 

L.  Tang,  i    d. 

L.  Cotg. 

L.  Cos. 

30 

8.78  568 

206 

8.78  649 

I .21  35i 

9.99919 

30 

206 

3i 

8.78  774 

205 

8.78855 

1 .21  145 

9.99918 

29 

32 

8.78979 

£.79  061 

I .20  939 

9.99917 

28 

33 

8.79  i83 

203 

8.79  266 

204 

I .20  734 

9.99917 

27 

34 

8.79  386 

8.79470 

1 .20  53o 

9.99916 

26 

35 

8.79  588 

8.79673 

I .20  327 

9.99915 

25 

36 

8.79  789 

201 

8.79875 

201 

1 .20  125 

9.99914 

24 

■^7 

8.79990 

199 

8.80076 

1 .19  924 

9.99913 

23 

38 

8.80  i«y 

8.80  277 

1 .  19  723 

9.99913 

22 

39 

8.80  388 

199 
197 
197 
196 

8.80476 

199 
198 

198 
196 
196 

1 .  19  524 

9.99912 

21 

40 

8.80  585 

8.80674 

1 .  19  326 

9.99911 

20 

4i 

8.80  782 

8.80872 

1 .19  128 

9.99910 

19 

42 

8.80978 

8.81  068 

1.18  932 

9.99909 

18 

43 

8.81  173 

8.81  264 

1. 18736 

9.99909 

17 

194 

195 

44 

8.81  367 

8.81  459 

i.i854i 

9.99908 

16 

45 

8.81  56o 

8.81  653 

1. 18  347 

9.99907 

i5 

46 

8.81  752 

192 

8.81  846 

193 

I. 18  i54 

9.99906 

i4 

47 

8.81  944 

8.82  o38 

192 

1 .  17  962 

9.99905 

i3 

48 

8.82  1 34 

8.82  23o 

1. 17  770 

9-99904 

12 

49 

8.82  324 

190 
189 

8.82  420 

190 

I  .  17  58o 

9-99904 

1 1 

50 

8.82  5i3 

188 
187 
187 
186 

8.82  610 

189 
188 

1.17  390 

9.99903 

10 

5i 

8.82  701 

8.82  799 

1 .  17  201 

9.99902 

9 

52 

8.82888 

8.82987 

188 

1 .  17  oi3 

9.99901 

8 

53 

8.83  075 

8.83  175 

186 

1.16825 

9.99900 

7 

54 

8.83261 

185 
184 
183 

8.83  361 

186 

1 .  16  639 

9.99899 

6 

55 

8.83  446 

8.83  547 

185 
184 

1.16453 

9.99898 

5 

56 

8.83  63o 

8.83732 

1. 16  268 

9.99898 

4 

57 

8.838i3 

183 
181 

iSi 

8.83916 

184 
182 

1.16084 

9.99897 

3 

58 

8.83996 

8.84  100 

I  .i5  900 

9.99896 

2 

59 

8.84  177 

8.84  282 

i.i5  718 

9.99895 

I 

60 

8.84  358 

8.84  464 

i.i5  536 

9.99894 

0 

L.  Cos.    !    d. 

L.  Cotg.       d. 

L.  Tang. 

L.  Sin. 

t 

86°. 

1 

PP 

.1 

201 

198 

195 

.1 

193 

189 

187 

.1 

185 

183 

181 

20.1 

19.8 

19- 5 

19.2 

18.0 

18.7 

,8.5 

18.3 

18.1 

.2 

40.2 

39.6 

390 

.2 

38.4 

37.8 

37-4 

.2 

37-0 

36.6 

36.2 

•3 

60.3 

59-4 

585 

•3 

57-6 

56-7 

56.1 

•3 

55-5 

54-9 

54-3 

•4 

80.4 

79.2 

78.0 

•4 

76.8 

75-6 

74.8 

•4 

74.0 

73.2 

72.4 

•S 

100.5 

gg.o 

97-5 

•5 

96.0 

94-.') 

93-5 

•5 

92-5 

91-5 

92- 1 

.6 

120.6 

118.8 

117. 0 

.6     115.2 

"3-4 

1 12.  2 

.6 

III.O 

109.8 

108.6 

•7 
.8 

140.7 

138.6 

»36-S 

■7      134-4 

132-3 

130.9 

•7 

129.5 

128.1 

126.7 

160.8 

158.4 

150.0 

.8     153.6 

151.2 

149.6 

.8 

148.0 

146.4 

144.8 

180.9  1  178.2  '  I7S.  5       1 

.g      172.8      170.1  1  168.3 

166.5  1  164.7  1  162.9     1 

il 


4 

3 

/ 

L.  Sin. 

d. 

L.  Tang.  !    d. 

L.  Cotg. 

L.  Cos. 

0 

8.84  358 

181 

8.84  464 

182 

tSo 

1 . 1 

5  536 

9.99894 

60 

I 

8.84539 

8.84  646 

5  354 

9- 

99  893 

59 

2 

8.84718 

8.84826 

5  174 

9- 

99  892 

58 

3 

8.84897 

179 
178 

8.85  006 

4  994 

9- 

99891 

^7 

4 

8.85  075 

8.85  185 

179 
178 

48i5 

9- 

99891 

56 

5 

8.85  252 

8.85  363 

4  637 

9- 

99  890 

DD 

6 

8.85429 

177 
.76 

8.85  540 

177 

4460 

9- 

99  889 

54 

7 

8.85  605 

8.85  717 

176 

4283 

9- 

99  888 

53 

8 

8.85  780 

8.85893 

4  107 

9- 

99887 

52 

9 

8.85  955 

«75 
173 

8.86  069 

176 
j'74 

3  931 

9- 

99886 

Dl 

10 

8.86  128 

8,86243 

3757 

9- 

99885 

50 

1 1 

8.86  3oi 

'73 
173 

8.86417 

'74 
'74 

3  583 

9- 

99884 

49 

12 

8.86  474 

8.86591 

3  409 

9- 

99  883 

48 

i3 

8.86  645 

171 

8.86763 

172 

3237 

9- 

99882 

47 

i4 

8.86816 

171 
171 

169 
169 

8.86935 

172 
171 

3065 

9- 

99881 

46 

i5 

8.86987 

8.87  106 

2894 

9- 

99880 

45 

i6 

8.87  i56 

8.87  277 

171 

2  723 

9- 

99879 

44 

n 

8.87  325 

169 

8.87447 

170 
169 

2  553 

9- 

99879 

43 

i8 

8.87494 

i67 
168 

8.87616 

2  384 

9- 

99878 

42 

•9 

8.87661 

8.87785 

169 

12  2l5 

9- 

99877 

4i 

20 

8.87  829 

166 
166 

8.87953 

.67 
.67 

12  047 

9- 

99876 

40 

21 

8.87995 

8.88  120 

11   880 

9- 

99875 

39 

22 

8.88  161 

165 
164 

8.88287 

I  I   713 

9- 

99874 

38 

23 

8.88  326 

8.88  453 

165 

11   547 

9- 

99873 

37 

24 

8.88  490 

164 

8.88618 

i6s 
'6S 

II  382 

9- 

99872 

36 

25 

8.88  654 

163 

8.88783 

I  217 

9- 

99871 

35 

26 

8.88817 

8.88948 

[I  o52 

9- 

99870 

34 

163 

'63 

27 

8.88980 

162 

8.89  III 

'63 

[O  889 

9- 

99  869 

33 

28 

8.89  142 

8.89  274 

10  726 

9- 

99868 

32 

29 

8.89  3o4 

8.89437 

163 

10  563 

9- 

99867 

3i 

30 

8.89464 

8.89598 

10  4o2 

9- 

99866 

30 

L.  Cos.        d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

85°  30 . 

PP 

181 

179 

177 

,1 

175 

173 

171 

.1 

168 

166 

164 

18. 1 

17.9 

177 

17-5 

17-3 

17.1 

16.  £ 

16.6 

16.4 

.2 

36.2 

35-8 

35-4 

.2 

35° 

34.6 

34a 

.2 

33- 1 

33-2 

32.8 

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54-3 

53-7 

53  I 

•3 

52-5 

5'-9 

51-3 

•3 

50.4 

49.8 

49.2 

•4 

724 

7..6 

70.8 

•4 

70.0 

69.2 

68.4 

•4 

67.= 

66.4 

656 

•5 

90-5 

89s 

88.5 

•5 

87.5 

86.5 

85.5 

•S 

84.C 

>      83.0 

82.0 

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108.6 

107.4 

[o6.a 

.6 

105.0 

103.8 

t02.6 

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100.  £ 

99.6 

98.4 

.7 

126.7 

1253 

123.9 

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122.5 

121. 1 

II9.7 

•7 

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116.2 

1 14. 8 

.8 

144-8 

1432 

141. 6 

.8 

140.0 

138.4 

136.8 

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132.8 

131.2 

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162.9  '  161. 1  1  159.3      1 

■9  '  »57-5  '  '55-7  I  '53-9 

38 


40 

30. 

/ 

L.  Sin.    !    d. 

L.  Tang. 

d.    1   L.Cotg. 

L.  Cos. 

30 

8.89  464 

161 

8.89  598 

162 
160 

1 .  10  4o2 

9.99  866 

30 

3i 

8 

89  625 

8.89  760 

1 .  10  240 

9 

99865 

29 

32 

8 

89784 

8.89  920 

1 .  10  080 

9 

99864 

28 

33 

8 

89943 

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159 
158 

8.90  080 

1 .09  920 

9 

99863 

27 

34 

8 

90   102 

8.90  24o 

1 .09  760 

9 

99  862 

26 

35 

8 

90  260 

8.90  399 

159 

1 .09  601 

9 

99  861 

25 

36 

8 

90  417 

157 
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8.90557 

158 
158 

1 .09  443 

9 

99  860 

24 

■^7 

8 

90  574 

156 

8.90  715 

1 .09  285 

9- 

99  859 

23 

38 

8 

90  730 

8.90  872 

157 

1 .09  128 

9- 

99  858 

22 

39 

8 

90885 

155 

8.91  029 

157 
156 

1 .08  971 

9- 

99857 

21 

155 

4U 

8 

91  o4o 

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8.91  185 

155 

i.o88i5 

9- 

99  856 

20 

4i 

8. 

91 195 

'54 

8.91  34o 

15s 

1 .08  660 

9- 

99855 

19 

42 

8 

91  349 

8.91  495 

1 .08  505 

9- 

99854 

18 

43 

8 

91  5o2 

153 
153 

8.91  650 

155 
153 

1.08  35o 

9- 

99853 

17 

44 

8. 

91  655 

8.91  8o3 

1 .08  197 

9- 

99  852 

16 

45 

8. 

91  807 

8.91  957 

1.08  043 

9- 

99  85i 

i5 

46 

8. 

91  959 

151 

8.92  110 

153 
152 

1 .07  890 

9- 

99  850 

i4 

47 

8. 

92  no 

8.92  262 

1.07  738 

9- 

99848 

i3 

48 

8. 

92  261 

8.92  4i4 

1.07  586 

9- 

99  847 

12 

49 

8. 

92  4ii 

150 
150 

8.92  565 

151 
151 

1.07  435 

9- 

99846 

1 1 

50 

8. 

92  56i 

8.92  716 

1 .07  284 

9- 

99845 

10 

5i 

8. 

92  710 

149 
149 
148 
147 

8.92866 

150 

1 .07  i34 

9- 

99844 

9 

52 

8. 

92  859 

8.93  016 

1 .06  984 

9- 

99843 

8 

53 

8. 

93  007 

8.93  165 

148 

1.06  835 

9- 

99  842 

7 

54 

8. 

93  i54 

8.93  3i3 

149 

1.06  687 

9- 

99  84i 

6 

55 

8. 

93  3oi 

8.93462 

1.06  538 

9- 

99  84o 

5 

56 

8. 

93448 

147 
146 
146 

8.93609 

147 
147 

147 
146 
146 

1 .06  391 

9- 

99  839 

4 

5? 

8. 

93  594 

8.93  756 

1.06  244 

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99  838 

3 

58 

8. 

93  740 

8.93  903 

1 .06  097 

9- 

99837 

2 

59 

8. 

93  885 

145 
MS 

8.94  049 

I .o5  951 

9- 

99  836 

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60 

8. 

94  o3o 

8.94  195 

I .o5  805 

9- 

99  834 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin.    j 

/ 

85°.                                                     1 

PP 

.1 

163 

160 

159 

.1 

157 

155 

153 

151 

149 

147 

16. 

i     16.0 

15-9 

'5-7 

I5-5 

iS-3 

.1 

15- 1 

14.9 

14.7 

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32-' 

i     32.0 

31.8 

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31-4 

31.0 

30.6 

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30.2 

29.8 

294 

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48. 

J     48.0 

47-7 

■3 

47-1 

4b- 5 

45-9 

•3 

45-3 

44-7 

441 

•4 

64. 

3      64.0 

63.6 

•4 

62.8 

62.0 

61.2 

•4 

60.4 

59-6 

58.8 

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81. 

3         80.0 

79-5 

•5 

78.5 

77-5 

76s 

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7.';-5 

74-5 

73-5 

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97- 

2         96.0 

95-4 

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94.2 

93.0 

91.8 

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90.6 

89.4 

88.2 

•  7 

113- 

{       II2.0 

111.3 

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109.9 

108.5 

107. 1 

.7 

I0.5-7 

104.3 

102.9 

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129. 

5     128.0 

127.2 

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125.6 

124  0 

122.4 

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120.8 

1 19. 2 

117.6 

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3     144.0 

'39-5 

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39 


8P, 


/ 

L.  Sin.   ;    d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

60 

0 

8 

94  o3o 

144 

143 
144 
142 

143 
141 
142 
141 
140 
140 

139 
139 
139 
138 

138 
137 

■37 
136 
136 
.36 

135 
13s 
134 
134 
"33 
133 
133 
132 
132 
131 

8 

94  195 

MS 
145 
145 
143 
144 
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142 
142 
142 
141 
140 
141 
139 
140 
138 
139 
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137 
138 
136 

137 
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134 
134 
133 
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133 

I  .o5  805 

9 

99  834 

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2 

3 

4 
5 
6 

7 

8 

9 

8 
8 
8 

8 
8 
8 

8 

8 
8 

94  174 

94  3i7 
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94  6o3 

94  746 
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95  029 
95  170 
95  3io 

8 
8 
8 

8 
8 
8 

8 
8 
8 

94  340 
94  485 

94  630 

94773 

94917 

95  060 

95  202 
95344 
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I  .o5  660 
i.o5  515 
I  .o5  370 

I  .o5  227 
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I  .o4  940 

1  .o4  798 
I.04  656 
I .04  5i4 

9 
9 
9 

9 
9 
9 

9 
9 

9 

99833 
99  832 
99831 

99  83o 
99829 
99  828 

99827 
99  825 
99  824 

59 
58 
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56 

55 
54 
53 

52 

5i 

10 

II 

12 

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i4 
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17 

i8 

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8 

95  450 

8 

95  627 

i.o4  373 

9 

99  823 

50 

8 
8 
8 

8 
8 
8 

8 

8 
8 

95  589 

95  728 
95867 

96  oo5 
96  i43 
96  280 

96  417 
96553 
96  689 

8 
8 
8 

8 
8 
8 

8 
8 
8 

95  767 

95  908 

96  047 

96  187 
96  325 
96464 

96  602 
96739 
96877 

1.04233 
1 .04  092 
i.o3  953 

i.o3  8i3 
i.o3  675 
I.03  536 

i.o3  398 
I  .o3  261 
i.o3  123 

9 
9 
9 

9 
9 
9 

9 
9 
9 

99  822 
99  821 
99  820 

99819 
99817 
99  816 

99  8i5 
99  8i4 
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49 
48 

47 
46 
45 
44 

43 
42 
4i 

20 

8 

96825 

8 

97  oi3 

1 .02  987 

9 

99812 

40 

21 

22 
23 

24 
25 
26 

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28 
29 

8 
8 
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8. 
8. 
8, 

8. 
8. 
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96  960 

97095 

97  229 

97  363 
97  496 
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97  762 

97  894 

98  026 

8 

8. 

8 

8. 
8. 
8. 

8. 
8. 
8. 

97  150 
97  285 

97  421 

97  556 
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97825 

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98  092 

98  225 

1 .02  85o 
1.02  715 
1 .02  579 

1 .  02  444 
1 .02  309 
1 .02  175 

1 .02  o4i 
1 .01  908 
i.oi  775 

9 

9- 

9- 

9- 
9- 
9- 

9- 
9- 
9- 

99  810 
99  809 
99  808 

99807 
99  806 
99  8o4 

99  8o3 
99  802 
99  801 

39 
38 
37 

36 
35 
34 
33 

32 

3i 

30 

8. 

98  i57 

8. 

98353 

1 .01  642 

9- 

99  800 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

' 

84°  30 .                                                    1 

PP 

.1 
.2 
■3 

•4 

■5 

.6 

•7 
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143 

141 

. 

.1 
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139 

138 

136 

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133 

131 

14  = 
29.  c 

43-5 

58.C 
72.5 
87.  c 

101.5 
116.C 

'30-5 

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28.6 
42.9 

57-2 
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85.8 

100.1 
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128.7 

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28.2 
423 

56.4 

84.6 

98.7 
112.8 

126.9 

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55-6 
69.5 

834 

97-3 
III. 2 

13-8 
27.6 
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55-2 
69.0 
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96.6 
10.4 
24.2 

136 
27.2 
40.8 

54- 4 
68.0 
81.6 

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13s 

27.0 
40-5 

S4.0 
67.5 
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108.0 
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39-9 

532 
66.5 
79.8 

100.4 

13- 1 
26.2 

39-3 

S2-4 
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78.6 

91.7 
104.8 

4o 


6°  30 . 


/ 

L.Siu.    1    d. 

L.  Tang.  \    d.      L.  Cotg. 

L.  Cos.           1 

30 

8.98  iSy 

8.98  358 

132 
132 

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9.99  800 

30 

3i 

8. 

98  288 

131 
131 

8.98490 

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9- 

99  798 

29 

32 

8 

98  419 

8.98622 

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99  797 

28 

33 

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130 
129 

8.98  753 

131 
131 

1 .01  247 

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34 

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8.98884 

1 .01  1 16 

9- 

99  795 

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8.99  015 

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36 

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8.99  145 

130 
130 

1 .00  855 

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99792 

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8.99  275 

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127 

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1 .00  595 

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99790 

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39 

8 

99  822 

8.99  534 

128 
129 

1 .00  466 

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99788 

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40 

8 

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8.99  662 

1. 00  338 

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99  787 

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8.99791 

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0.99  954 

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126 

0.99  826 

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45 

9 

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0.99699 

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46 

9 

00  207 

9.00  427 

0.99  573 

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99  765 

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0.98  082 
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9 
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99  764 
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9.02  162 

0.97838 

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99  761 

0 

L.  Cos.        d. 

L.  Cotg.  1    d.    1  L.  Tang. 

L.  Sin. 

/ 

84°. 

PP 

.1 

130 

1 
129 

128 

.1 

126 

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123 

.1 

122 

121 

120 

13.0 

12.9 

12-8 

12.6 

12.  S 

12.3 

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12.0 

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26.0 

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25.6 

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25.2 

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24.6 

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375 

369 

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51.6 

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50.0 

49.2 

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3      48.4 

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65.0 

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62.5 

61.S 

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61. c 

3      60.5 

60.0 

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78.0 

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76.8 

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75.0 

73-8 

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87.5 

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t      84.7 

84.0 

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104.0 

103.2 

102.4 

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100.8 

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98.4 

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5      96.8 

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I16.I       I15.2        1             .Q    1    I13.4   1    I12.5   1    IIO.7 

log.  8  1  108.9  1 

108.0    1 

4r 


6°. 


10 


20 


22 
23 

24 
25 
26 

27 
28 

30 


L.  Sin. 

9.01  923 

9.02  o43 
9.02  i63 
9.02  283 

9.02  4o2 

9.02  520 

9.02  639 

9,02  767 
9.02  874 
9.02  992 


9.03  109 


9.03  226 
9.03  342 
9.03458 

9.o3  574 
9.03  690 
9.03  805 

9.03  920 

9.04  o34 
9.04  149 


9.04  262 


9.04  376 
9.04  490 
9.04  6o3 

9.04  71 5 
9.04828 

9.04  940 

9.05  o52 
9.05  164 
9.05  275 


9.05  386 


L.  Cos. 


d. 

120 
1 30 
120 
119 
118 
119 
118 

"7 
118 
117 

"7 
116 
116 
116 
116 
"5 
"5 
114 

"5 
113 

114 
114 

"3 
112 

"3 
112 
112 

112 
III 
III 


d. 


L.  Tang. 

9.02  162 

9.02  283 
9.02  4o4 
9.02  525 

9.02  645 
9.02  766 

9.02  885 

9.08  005 

9.03  124 
9.03  242 


.o3  36 1 


9.03  479 
9.03  697 
9.03  714 

9.03  832 

9.03  948 

9.04  o65 

9.04  181 
9.04  297 
9 . o4  4 1 3 


9.04  528 


9.04  643 
9.04  758 
9.04  873 

9.04  987 

9.05  lOI 

9.06  2l4 

9.05  828 

9.05  44i 
9.05  553 


9.05  666 


d.    I  L.  Cotg. 

0.97  bjid 


121 

131 
131 

I20 
121 
119 
120 
119 

1x8 
119 
118 
118 
117 
118 
116 
117 
116 
116 
116 
"5 
"S 
"5 
"S 
114 
114 
"3 
114 
"3 
112 
"3 


L.  Cotg.  I  d. 


0.97  717 
0.97  596 
0-97475 
0.97  355 
0.97  234 
0.97  115 

0.96  995 
0.96  876 
0.96  758 


.96  639 


0.96  521 

0.96  4o3 
0,96  286 

0.96  168 
0.96  o52 
0.95  935 

0.95  819 
0.95  703 
0.95  587 


.95  472 


0.95  357 
0.95  242 
0.95  127 

0.95  oi3 
0.94  899 
0.94  786 

0.94  672 
0.94  559 
0.94  447 


0,94  334 


L.  Tang. 


L.  Cos. 

9-99  761 

99  760 
99759 
99757 

99  756 

99  7^5 
99753 

99752 
99751 
99  749 


99  748 


99  747 
99  745 
99  744 

99  742 
99741 
99  74o 

99  738 
99787 
99  736 


99  734 


99733 
99781 
99780 

99  728 
99727 
99  726 

99  724 
99728 
99  721 


9972 


L.  Sin.  I 


83 

°30 

• 

pp 

121 

130 

119 

.1 

118 

X17 

116 

.1 

"5 

114 

"3 

.1 

12. 1 

12.0 

II.9 

11.8 

11.7 

11.6 

"5 

II. 4 

"3 

.2 

•3 

24.2 
36.3 

24.0 
36.0 

23.8 

35-7 

2 
•3 

23.6 

35-4 

23.4 

35- 1 

23.2 
34-8 

.2 

•3 

23.0 
34-5 

22.8 
34-2 

22.6 
33-9 

•4 
•5 
.6 

60.5 
72.6 

48.0 
60.0 
72.0 

47.6 
59- S 
71.4 

•4 
•S 
.6 

47.2 
S9.0 
70.8 

46.8 
58.5 
70.2 

46.4 
58.0 
69.6 

•4 
•5 
.6 

46.0 

69.0 

45.6 
57.0 
68.4 

56.5 
67.8 

•7 

8 

84.7 
96.8 
108.9 

84.0 
96.0 
108.0 

83.3 
95.2 

•7 
.8 

83.6 

106. 3 

81.9 
93.6 

8r.2 
92.8 

-.1 

80.5 
92.0 

79.8 

91.2 
J02.6 

79.1 
90.4 
101.7 

42 


6°  30 . 

/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

30 

9.05  3ts6 

m 

9.06  666 

112 

0.94  334 

9.99  720 

30 

3i 

9.05  497 

9.05  778 

0.94  222 

9- 

99  718 

29 

32 

9.05  607 

9.06  890 

0.94  1 10 

9- 

99  717 

28 

33 

9.05  717 

no 

9.06  002 

III 

0.93  998 

9- 

99716 

27 

34 

9.05  827 

9.06  1 13 

0.93  887 

9- 

99714 

26 

35 

9.05  937 

9.06  224 

0.93  776 

9- 

99  7'3 

25 

36 

9.06  o46 

109 
109 

9.06  335 

no 

0.93665 

9- 

99  711 

24 

^7 

9.06  i55 

9.06  445 

0.93  555 

9- 

99710 

23 

38 

9.06  264 

9.06  556 

0.93444 

9- 

99  708 

22 

39 

9.06  372 

108 
109 

9.06  666 

no 
109 

0.93  334 

9- 

99  707 

21 

40 

9.06  4B1 

9.06  775 

0.93  225 

9- 

99705 

20 

4i 

9.06  589 

107 
108 
107 

9.06885 

109 

0.93  ii5 

9- 

99  704 

J9 

42 

9.06  696 

9.06  994 

0.93  006 

9- 

99  702 

18 

43 

9.06  8o4 

9.07  io3 

109 

0.92  897 

9- 

99701 

17 

44 

9.06  91 1 

9.07  211 

0.92  789 

9- 

99699 

16 

45 

9.07  018 

106 

9.07  320 

.108 

0.92  680 

9- 

99  698 

i5 

46 

9.07  124 

9.07  428 

0.92  572 

9- 

99  696 

i4 

107 

108 

47 

9.07  23l 

106 

9.07  536 

0.92  464 

9- 

99695 

i3 

48 

9.07  337 

9.07  643 

108 

0.92  357 

9- 

99  693 

12 

49 

9.07  442 

105 
106 

9.07  761 

0.92  249 

9- 

99  692 

II 

107 

50 

9.07  548 

9.07  858 

106 

0.92  142 

9- 

99  690 

10 

bi 

9.07  653 

9.07  964 

0.92  o36 

9- 

99  689 

9 

62 

9.07  768 

9.08  071 

106 
106 

0.91  929 

9- 

99  687 

8 

63 

9.07863 

105 
105 

9.08  177 

0.91  823 

9- 

99686 

7 

b4 

9.07  968 

104 

9.08  283 

106 

0.91  717 

9- 

99684 

6 

55 

9.08  072 

9.08  389 

106 

0.91  611 

9- 

99683 

5 

56 

9.08  176 

9.08495 

0.91  5o5 

9- 

99  681 

4 

104 

'OS 

i57 

9.08  280 

9.08  600 

0.91  4oo 

9- 

99  680 

3 

58 

9.08  383 

9.08  7o5 

0.91  295 

9- 

99  678 

2 

59 

9.08486 

103 

9.08  810 

105 

0.91  190 

9- 

99677 

I 

60 

9.08  589 

9.08  914 

0.91  086 

9- 

99675 

0 

L.  Cos. 

d. 

L.  Cotg. 

1    d. 

L.  Tang. 

L.  Sin. 

/ 

83°. 

1 

PP 

.1 

112 

III 

no 

.1 

109 

108 

10.8 

107 
10.7 

.1 

106 

105 

104 

II. 2 

II. I 

II. 0 

10.9 

10.6 

10.5 

10.4 

.2 

22.4 

22.2 

22.0 

.2 

21.8 

21.6 

21.4 

.2 

21.2 

21.0 

20.8 

3 

33.6 

33-3 

33.0 

•3 

32.7 

32.4 

32.1 

3 

31k 

3I-S 

31.2 

•4 

44.8 

44.4 

44.0 

•4 

436 

43-2 

42.8 

•4 

42.4 

42.0 

41.6 

•5 

56.0 

55-5 

550 

5 

54-5 

.S4-0 

535 

5 

.■;3-c 

525 

52.0 

.6 

67.2 

66.6 

66.0 

.6 

65.4 

64.8 

64.2 

.6 

63.4 

63.0 

62.4 

•7 

78.4 

77-7 

77.0 

.7 

763 

7S.6 

74-9 

•7 

74.3 

735 

72.8 

8 

89.6 

88.8 

88.0 

.8 

87.2 

864 

85.6 

.8 

84.!: 

84.0 

83.2 

-^ 

100.8  !     gg.g  1     99.0      1 

9  1    98.1  1    97.2  1    96.3 

95.4  1    94.5  !     93.6    1 

43 


7°. 

t 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

0 

9 .  08  589 

103 
103 

9.08  914 

105 
104 

0.91  086 

9.99675 

60 

I 

2 

9- 
9- 

08  692 
08795 

9.09  019 

9.09  123 

0.90  981 
0.90  877 

9- 
9- 

99674 
99672 

59 

58 

3 

9- 

08  897 

1 03 

9.09  227 

104 
103 

0.90  773 

9- 

99  670 

^7 

4 

9- 

08  999 

9.09  33o 

104 

0.90  670 

9- 

99  669 

56 

b 

9- 

09   lOI 

9.09  434 

0.90  566 

9- 

99667 

3b 

6 

9- 

09  202 

102 

9.09  537 

103 
103 

0.90  463 

9- 

99  666 

54 

7 

9- 

09  3o4 

9.09  64o 

0.90  36o 

9- 

99  664 

53 

8 

9- 

09405 

9.09  742 

0.90  258 

9- 

99  663 

52 

9 

9- 

09  5o6 

9.09  845 

103 

0.90  i55 

9- 

99  661 

5i 

10 

9- 

09  606 

9.09947 

0.90  o53 

9- 

99  659 

50 

1 1 

9- 

09  707 

9.10  049 

0.89  95i 

9- 

99  658 

49 

12 

9- 

09  807 

9.10  i5o 

0.89  850 

9- 

99  656 

48 

i3 

9- 

09907 

•99 

9. 10  262 

lOt 

0.89  748 

9- 

99  655 

47 

i4 

9- 

TO  006 

9.10353 

0.89  647 

9- 

99  653 

46 

13 

9- 

10   106 

9.10454 

0.89  546 

9- 

99  65i 

45 

i6 

9- 

10  205 

99 
99 

9.10555 

lOI 

0.89445 

9- 

99650 

44 

17 

9- 

10  3o4 

98 

9. 10  656 

0.89  344 

9- 

99648 

43 

i8 

9- 

10  402 

9. 10  756 

0.89  244 

9- 

99  647 

42 

'9 

9- 

10  5oi 

99 
98 

98 
98 

9.I0856 

0.89  i44 

9- 

99645 

4i 

20 

9- 

10  599 

9.10  956 

0.89  o44 

9- 

99  643 

40 

21 

9- 

10  697 

9. 1 1  o56 

99 

0.88944 

9- 

99  642 

39 

22 

9- 

10  795 

98 
97 

9. 1 1  i55 

0.88845 

9- 

99  64o 

38 

•zi 

9- 

10  893 

9.  II  254 

99 
99 

0.88  746 

9- 

99  638 

37 

24 

9- 

10  990 

9. II  353 

0.88  647 

9- 

99637 

36 

2l) 

9- 

1 1  087 

9. 1 1  452 

0.88  548 

9- 

99  635 

35 

26 

9- 

11  i84 

97 
97 

9. 1 1  55i 

99 
98 

0.88449 

9- 

99  633 

34 

27 

9- 

II  281 

96 

9. 1 1  649 

98 
98 
98 

0.88  35i 

9- 

99  632 

33 

28 

9- 

11  377 

9- 1 1  747 

0.88253 

9- 

99  63o 

32 

29 

9- 

1 1  474 

97 

96 

9. 1 1  845 

0.88  155 

9- 

99  629 

3i 

30 

9- 

1 1  670 

9. 1 1  943 

0.88  057 

9- 

99  627 

30 

L.  Cos. 

d. 

L.  Cotg.  1    d.      L.  Tang. 

L.  Sin. 

' 

82°  30 .                                                  1 

PP 

105 

104 

103 

.1 

103 

101 

lOO 

99 

98 

97 

.1 

10.5 

10.4 

10.3 

10.2 

10. 1 

10. 0 

.1 

9.9 

9.8 

9  7 

.2 

21. 0 

20.8 

20  6 

.3 

20.4 

20.2 

20.0 

.2 

198 

19.6 

19.4 

•3 

3'  5 

3>-2 

30.9 

•3 

30.6 

30-3 

30.0 

3 

29.7 

29.4 

29.1 

•4 

42.0 

41.6 

41.2 

4 

40.8 

40.4 

40.0 

•4 

39.6 

392 

38.8 

•5 

S2.S 

52.0 

5I-5 

•5 

51.0 

So-S 

50.0 

■5 

49-5 

49.0 

48.5 

.6 

63.0 

62.4 

6i.8 

.6 

61.2 

60.6 

60.0 

.6 

59  4 

58.8 

58.2 

.7 

71- ■; 

72.8 

72.1 

•7 

71.4 

70.7 

7a  0 

.7 

69.3 

68.6 

67.9 

8 

«4.o 

83.2 

82.4 

.8 

81.6 

80.8 

80.0 

.8 

79.2 

78.4 

77.6 

.9      91.8  1    00.9 

90.0 

8c. 

88.2 

87^ 

44 


30 

3i 

32 

33 

34 
35 
36 

37 
38 
39 


40 


4i 

42 

43 

44 
45 
46 

47 

48 

49 


50 


60 


L.  Sin. 

I  570 


I  666 
I  761 

1  857 

1  952 

2  o47 
2  l42 

2  236 
2  33i 

2425 


2  519 


2  612 
2  706 
2  799 

2  892 

2  985 

3  078 

3  171 
3  263 
3  355 


3447 


3  539 
3  63o 
3  722 

38i3 
3  904 

3  994 

4o85 
4175 

4  266 


4  356 


d. 


7°  30 . 

L.  Tang.  1  d. 


1  943 

2  o4o 
2  i38 
2  235 

2  332 
2428 
2  525 

2  621 
2  717 
2  8i3 


909 


3  oo4 
3  099 
3  194 

3  289 
3  384 
3478 

3573 
3667 
3  761 


3  854 


3948 
4  o4i 
4i34 

4  227 
4  320 
4412 

4  5o4 
4  597 

4  688 


4  780 


97 
97 
96 

97 
96 

96 
96 
96 

95 
95 
95 
95 
95 
94 
95 
94 
94 
93 

94 
93 
93 
93 

93 
92 
92 
93 
91 
92 


L.  Cotg. 

0.88  057 

0.87  960 
0.87  862 
0.87  765 

0.87  668 
0.87  572 
0.87475 

0.87  379 
0.87  283 
0.87  187 


0.87  091 


0.86  996 
0.86  901 
0.86  806 

0.86  711 
0.86  616 

0.86  522 
0.86  427 

0.86  333 
0.86  239 


0.86  i46 


0.86  o52 
0.85  959 
o. 85  866 

0.85  773 
0.85  680 
0.85  588 

0.85  496 
0.85  4o3 
0.85  3i2 


0.85  220 


L.  Cos. 

9-99627 

99  625 
99  624 
99  622 

99  620 
99  618 
99617 

99  6i5 
99  6i3 
99  612 


99  610 


99  608 
99  607 
99  605 

99  6o3 
99  601 
99  600 

99  598 
99  596 
99595 


99  593 


99  591 
99  589 
99  588 

99  586 
99  584 
99  582 

99  58i 
99  579 

99  577 


9.99  575 


L.  Cos. 


d. 


L.  Cotg. 


d. 


L.  Tang. 


L.  Sin. 


82°. 


PP. 

97 

96 

95 

.  I 

•  a.. 

94 

93 

92 

.1 
.2 

91 

90 

.1 

.2 

9-7 
19.4 

9.6 
19.2 

9-5 
19.0 

9-4 
18.8 

9-3 
18.6 

9.2 
18.4 

^11 

9.0 
18.0 

•3 

29.1 

28.8 

28.  s 

•3 

28.2 

27.9 

27.6 

•3 

-a?- 3, 

..270 

•4 
•5 
.6 

38.8 
48- 5 
58.2 

38.4 
48.0 

57-6 

38.0 
47-5 
570 

•4 

•5 
.6 

37-6 
47.0 

56.4 

37-2 
46.  s 
55-8 

368 
46.0 
55-2 

•4 
■5 
.6 

36.4 
45-5 
54-6 

36.0 
450 
540 

■7 
.8 

67.9 
77.6 

67.2 
76.8 
86.4 

66.5 
76.0 
85.5 

•7 
8 

65.8 
75-2 
84.6 

65.1 
74-4 
83^ 

64.4 
73.6 
82.8 

■7 
.8 

637 
72.8 

63.0 
72.0 
81.0 

45 


8< 

3 

• 

/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

0 

9.14356 

89 
90 
89 

9.1 

4  780 

92 

9» 

0.85  220 

9.99575 

60 

I 

9.14445 

9.1 

4872 

0.85  128 

9 

99  574 

59 

2 

9.14535 

9.1 

4963 

0.85  o37 

9 

99572 

58 

3 

9.14  624 

9.1 

5o54 

9' 

0.84946 

9 

99570 

i>7 

4 

9.14  714 

90 
89 

9.1 

5  145 

91 

0. 84  855 

9 

99  568 

56 

5 

9.14803 

9.1 

5  236 

0.84  764 

9 

99  566 

55 

6 

9.14  891 

89 
89 

88 
88 
88 
88 

9.1 

5327 

9» 

0.84673 

9 

99565 

54 

7 

9.  i4  980 

9- 

5  417 

0.84  583 

9 

99  563 

53 

8 

9. 1 5  069 

9- 

5  5o8 

0.84  492 

9 

99  56i 

52 

9 

9.15  i57 

9- 

5  598 

90 
90 
89 

0.84  402 

9 

99  559 

5i 

10 

9.15245 

9- 

5  688 

0.84  3i2 

9 

.99557 

50 

1 1 

9.15  333 

9. 

5  777 

0.84  223 

9 

.99  556 

49 

12 

9.i5  421 

9- 

5867 

89 

0.84  i33 

9 

99  554 

48 

i3 

9.15  5o8 

87 
83 
87 
87 
87 

9- 

5956 

0.84  o44 

9 

.99  552 

47 

i4 

9. 1 5  596 

9- 

6  o46 

90 
89 

0.83954 

9 

.99  55o 

46 

i5 

9.15683 

9- 

6135 

0. 83  865 

9 

.99548 

4b 

i6 

9.15770 

9- 

6  224 

89 

0.83  776 

9 

.99  546 

44 

I? 

9.15  857 

87 

9- 

6  3i2 

89 
88 
88 
88 
88 

0. 83  688 

9 

.99545 

43 

i8 

9.15  944 

9- 

6  4oi 

0.83  599 

9 

.99  543 

42 

•9 

9.16  o3o 

86 

87 

86 

9- 

6489 

0.83  5ii 

9 

.99  54r 

4i 

20 

9. 16  116 

9- 

[6  577 

0. 83  423 

9 

.99  539 

40 

2  1 

9. 16  203 

9- 

6  665 

0. 83  335 

9 

.99537 

39 

22 

9. 16  289 

85 
86 

9- 

6753 

88 
87 
88 

0.83  247 

9 

.99  535 

38 

23 

9.16374 

9- 

[6  84i 

0.83  159 

9 

.99  533 

37 

24 

9.16  460 

85 

9- 

[6928 

0.83  072 

9 

.99  532 

36 

25 

9.16  545 

86 

9- 

[7  016 

87 

0.82984 

9 

.99  53o 

35 

26 

9. 16  63i 

9- 

[7  io3 

0.82897 

9 

.99  528 

34 

85 

87 

27 

9.16  716 

8s 

9- 

[7  190 

87 

0.82  810 

9 

.99  526 

33 

28 

9.16  801 

9- 

17277 

0.82  723 

9 

.99  524 

32 

29 

9.16886 

85 

9- 

17  363 

0.82  637 

9 

.99  522 

3 1 

30 

9.16970 

9.17  450 

0.82  55o 

9 

.99  520 

30 

L.  Cos. 

d. 

L.  cotgr. 

d. 

L.  Tangr. 

L.  Sin. 

81° 

30. 

1 

PP 

93 

91 

90 

89 

88 

87 

86 

.1 

9.2 

?•' 

9.0 

.1 

8.9 

8.8 

X 

8.7 

8.6 

.2 

18.4 

] 

8.2 

18.0 

.2 

17.8 

.7.6 

2 

17-4 

17.2 

•3 

27.6 

7-3 

27.0 

•3 

26.7 

26.4 

3 

26.1 

25.8 

•4 

36.8 

6.4 

36.0 

•4 

356 

352 

4 

34-8 

34-4 

•5 

46.0 

A 

SS 

45.0 

•5 

44-5 

44.0 

S 

43-5 

430 

.6 

55-2 

4.6 

540 

.6 

53-4 

52.8 

6 

52.2 

51.6 

•7 

64.4 

( 

1-7 

63.0 

•7 

62.3 

61.6 

7 

60.9 

60.2 

.8 

73-6 

•> 

2.8 

72.0 

.8 

71.2 

70.4 

8 

69.6 

68.8 

.9          82.8     1      i 

I.Q                81.0 

80.1 

^^^ 

46 


8°  30 . 

/ 

L.Sin.        d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

30 

9.16  970 

85 
84 
84 
84 

84 
83 
84 

83 

83 
83 

83 
83 
82 
82 

9- 

17450 

86 
86 
86 

o.«2  55o 

9.99  620 

30 

3i 

32 

33 

9 

9 
9 

.17055 
.17  139 

17  223 

9- 
9- 
9- 

17  536 
17  622 
17708 

0.82464 
0.82  378 
0.82  292 

9 
9 
9 

99  5i8 
99517 
99515 

29 

28 
27 

34 
35 
36 

9 
9 
9 

.17307 
.17391 
.17  474 

9- 
9- 
9- 

17  794 
17  880 
17965 

86 
85 

0.82  206 
0.82  120 
0.82  035 

9 
9 
9 

99  5i3 
99  5i  1 
99  509 

26 

25 

24 

3? 
38 
39 

9 

9 
9 

.17558 
.17641 
.17  724 

9- 
9- 

9- 

i8o5i 
18  i36 
18  221 

85 
85 
85 

85 
84 
85 
84 
84 
84 
84 

83 
84 
83 

83 
83 
83 
83 

0.81  949 
0.81  864 
0.81  779 

9 
9 
9 

99507 
99  5o5 
99  5o3 

23 
22 
21 

40 

9 

17  807 

9- 

18  3o6 

0.81  694 

9 

99  5oi 

20 

4i 

42 

43 

9 
9 
9 

17  890 
17973 

.i8o55 

9- 
9- 
9- 

18  391 
18475 
18  56o 

0.81  609 
0.81  525 
0.81  44o 

9 
9 
9 

99499 
99497 
99  495 

>9 

18 

17 

44 
45 
46 

9 
9 
9 

.18  137 
.  18  220 
.18  302 

83 
82 
81 
82 
82 
81 

81 
81 
81 
81 

9- 
9- 
9- 

18644 
18  728 
18  812 

0.81  356 
0.81  272 
0.81  188 

9 
9 
9 

99  494 
99  492 
99  490 

16 
i5 
i4 

47 
48 

49 

9 
9 
9 

.18  383 
.18465 
.18  547 

9- 
9- 
9- 

18896 
18979 
19  o63 

0.81  io4 
0.81  021 
0.80  937 

9 
9 

9 

99488 
99486 
99  484 

i3 
12 
1 1 

50 

9 

.18628 

9- 

19  i46 

0.80854 

9 

99  482 

10 

5i 

52 

53 

9 
9 
9 

.18  709 
.18790 

.18871 

9- 
9- 
9- 

19  229 
19  3l2 
19  395 

0.80  771 
0,80  688 
0.80  605 

9 
9 
9 

99  480 
99478 
.99  476 

9 

8 

7 

54 
55 
56 

9 
9 
9 

.18952 
.  19  o33 
.19  ii3 

81 
80 
80 

9- 
9- 
9- 

19478 
19561 
19643 

83 
82 
82 

0.80  522 
0.80439 
0.80  357 

9 
9 
9 

99  474 
.99  472 
.99  470 

6 

5 
4 

57 
58 
59 

9 
9 
9 

.19  193 
.19273 
.19353 

80 
80 
80 

9- 
9- 
9- 

19  725 
19807 
19  889 

82 
82 
82 

0.80  275 
0.80  193 
0.80   III 

9 
9 
9 

.99  468 
.99  466 
.99  464 

3 
2 

I 

60 

9 

.19433 

9- 

19  971 

0.80  029 

9 

.99  462 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

1    L.Sin. 

/ 

81 

1°. 

1 

PP 

.1 

.2 

•3 

86 

85 

84 

.1 

.2 

•3 

83 

82 

I 

.2 
•3 

81 

80 

8.6 
17.2 
25.8 

8.5 
17.0 

2S-5 

8.4 
16.8 
25.2 

8.3 
16.6 
24.9 

8.2 
16.4 
24.6 

8.1 
16.2 
243 

8.0 
16.0 
24.0 

•4 

34-4 
430 
S1.6 

340 
42.5 
510 

33.6 
42.0 
50.4 

■4 
•5 
.6 

33-2 
4I-S 
49.8 

32.8 
41.0 
49.2 

•4 

5 

.6 

32' 4 
40.5 
48.6 

32.0 
40.0 
48.0 

•  7 
.8 

60.2 
68.8 

59-5 
68.0 

76.5 

58.8 
67.2 

75-6 

■7 
.8 

58.1 
66.4 

lit 

'j^ 

567 
64.8 

56  0 
64.0 

7?.o 

47 


0 

°. 

1 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

0 

9.19  433 

80 

9.19971 

0.80  029 

9.99  462 

60 

82 

I 

9 

.19  5i3 

79 
80 

9.20  o53 

81 

0.79947 

9 

99  460 

D9 

2 

9 

.  19  692 

9.20  1 34 

82 

0,79  866 

9 

99  458 

58 

•i 

9 

.  19  672 

9.20  216 

0.79  784 

9 

99  456 

i>7 

79 

81 

4 

9 

.19751 

79 

9.20  297 

81 

0.79  708 

9 

99  454 

b6 

b 

9 

.  19  83o 

9.20  378 

0.79  622 

9 

99  452 

b5 

6 

9 

.19909 

79 
79 

9.20  459 

81 

0.79  54i 

9 

99450 

54 

7 

9 

.19  988 

79 
78 
78 

9.20  540 

81 

0.79  460 

9 

99  448 

b3 

8 

9 

.20  067 

9.20  621 

80 
81 

0.79379 

9 

99  446 

b2 

9 

9 

.20  i45 

9.20  701 

0.79299 

9 

99  444 

bi 

10 

9 

.20  223 

9.20  782 

80 
80 

0.79  218 

9 

99  442 

50 

1 1 

9 

,20  3(12 

79 
78 

9.20  862 

0.79  1 38 

9 

99  44o 

49 

12 

9 

.2o38o 

78 

9.20  942 

80 
80 

0.79  o58 

9 

99  438 

48 

i3 

9 

.20  458 

9.21  022 

0.78  978 

9 

99  436 

47 

i4 

9 

.20  535 

78- 

9.21  102 

80 

0.78898 

9 

99434 

46 

lb 

9 

.20  6i3 

78 
77 
77 

9.21  182 

0.78  818 

9 

99  432 

4b 

iG 

9 

.20  691 

9.21  261 

79 
80 

0.78789 

9 

99429 

44 

'7 

9 

.20  768 

9.21  34 1 

79 

0.78  559 

9 

99427 

48 

i8 

9 

.20845 

9.21  420 

0.78  58o 

9 

99  425 

42 

'9 

9 

.20  922 

77 

9.21  499 

79 

0.78  5oi 

9 

99  428 

4i 

20 

9 

.20  999 

9.21  578 

79 

0.78  422 

9 

.99  421 

40 

77 

79 

21 

9 

.21  076 

77 

9.21  657 

79 

78 

0.78  343 

9 

99419 

89 

22 

9 

.21  i53 

76 

9.21  736 

0.78  264 

9 

99417 

88 

•2.S 

9 

.21  229 

9.21  8i4 

0.78  186 

9 

99415 

37 

77 

79 

24 

9 

.21  3o6 

76 

9.21  893 

78 

0.78  107 

9 

99418 

36 

2b 

9 

.21  382 

76 
76 

9.21  971 

78 
78 

0.78  029 

9 

99  4ii 

3  b 

2(3 

9 

.21  458 

9.22  049 

0.77951 

9 

99  409 

34 

27 

9 

.21  534 

76 

9.22  127 

78 

0.77873 

9 

99  407 

88 

28 

9 

,21  610 

9.22  205 

78 
78 

0-77  795 

9 

99  4o4 

82 

29 

9 

.21  685 

76 

9.22  283 

0.77717 

9 

99  4o2 

81 
30 

30 

9 

.21  761 

9.22  36i 

0.77  689 

9 

99  4oo 

L.  Cos. 

d. 

L.  Cotgr. 

d. 

L.  Tang. 

L.  Sin. 

' 

80° 

30. 

1 

pp 

.1 

83 

81 

80 

.1 

79 

78 

I 

77 

7« 

8.2 

8.1 

8.0 

7-9 

7.8 

7-7 

7.6 

.2 

16.4 

] 

6.2 

16.0 

.3 

15.8 

15.6 

2 

154 

"5-2 

3 

24.6 

s 

4-3 

24.0 

•3 

23.7 

a3-4 

3 

23.1 

22.8 

•4 

32.8 

2-4 

32.0 

•4 

31.6 

31.2 

4 

308 

30-4 

•5 

41.0 

A 

o-S 

40.0 

■5 

39-5 

39° 

s 

38.5 

38.0 

.6 

49.2 

4 

8.6 

48.0 

,6 

47-4 

46.8 

6 

46.2 

45.6 

■I 

57-4 

1 

6.7 

56.0 

.7 

55-3 

54-6 

7 

53-9 

53-2 

65.6 

< 

.4.8 

64.0 

.8 

63.2 

62.4 

8 

61.6 

60.8 

2.g            72.0     1 

_68^ 

48 


9°  30 . 

30 

L.  Sin.        d.    1  L.  Tang. 

d. 

L.  Cotg.       L.  Cos. 

9.21  761 

75 
76 

9  .22   36l 

77 
78 

0.77  639 

9.99  4oo 

30 

3i 

9.21  836 

9.22  438 

0.77  562 

9.99  398 

29 

•62 

9.21  912 

9.22  5i6 

0.77  484 

9.99396 

28 

33 

9.21  987 

75 

9.22  593 

77 

0.77  407 

9.99394 

27 

34 

9.22  062 

75 

9.22  670 

0.77  33o 

9.99  392 

26 

3b 

9.22  137 

9.22  747 

0.77  253 

9.99390 

25 

36 

9.22  211 

75 

9.22  824 

77 

0.77  176 

9.99  388 

24 

37 

9.22  286 

9.22  901 

76 

0.77  099 

9.99  385 

23 

38 

9.22  36i 

9.22  977 

0.77  023 

9.99  383 

22 

39 

9.22  435 

74 
74 

74 

9.23  o54 

77 
76 

76 

0.76  946 

9.99  38i 

21 

40 

9.22  509 

9.23  i3o 

0.76  870 

9.99379 

20 

4i 

9.22  583 

9.23  206 

0.76  794 

9.99  377 

'9 

42 

9.22  557 

9.23  283 

76 
76 

75 
76 

0.76  717 

9.99875 

18 

43 

9.22  731 

74 

9.23  359 

0.76  64i 

9.99372 

17 

44 

9.22  805 

9.23  435 

0.76  565 

9.99870 

16 

45 

9.22  878 

9.23  5io 

0.76  490 

9.99868 

i5 

46 

9.22  952 

9.23  586 

0.76  4i4 

9.99  366 

i4 

73 

75 

47 

9.28  025 

9.23  661 

76 

0.76  339 

9.99  364 

i3 

48 

9.23  098 

9.23  737 

0.7B  263 

9.99  862 

12 

49 

9.23  171 

73 

9.23  812 

75 

0.76  188 

9.99359 

1 1 

50 

9.23  244 

73 

9.23  887 

75 
75 

0.76  1 13 

9.99357 

10 

5i 

9.23  317 

9.23  962 

0.76  o38 

9.99  355 

9 

52 

9.23  390 

9.24  087 

0.75  963 

9,99  858 

8 

53 

9.23  462 

73 

9.24  112 

74 

0.75888 

9.99  35i 

7 

54 

9.23  535 

72 

g.24  186 

75 

0.75  8i4 

9.99  848 

6 

55 

9.23  607 

9.24  261 

0.75  789 

9.99  846 

5 

56 

9.23  679 

73 

9.24  335 

74 
75 

0.75  665 

9.99  844 

4 

37 

9.23  752 

9.24  4io 

74 

0.75  590 

9-.  9  9  842 

8 

58 

9.23823 

9.24484 

0.75  5i6 

9.99  340 

2 

59 

9.23895 

9.24558 

74 

0.75  442 

9.99  887 

I 

60 

9.23  967 

9.24  632 

0.75  368 

9.99335 

0 

L.  Cos.    1    d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

/ 

8( 

)°. 

1 

PP 

77 

76 

75 

.1 

74 

73 

.1 

79 

71 

7-7 

7.6 

7-S 

^i 

7-3 

7.2 

7- 1 

.2 

iS-4 

15.2 

15.0 

.2 

14.8 

14.6 

.2 

14.4 

142 

•3 

23.1 

22.8 

22.5 

•3 

22.2 

21.9 

•3 

21.6 

2'-3 

•4 

30.8 

30-4 

30.0 

•4 

29.6 

29.2 

•4 

28.8 

28.4 

•  5 

38-5 

38.0 

37-5 

■  s 

37.0 

36. 5 

■S 

36.0 

35- S 

.6 

46.2 

45-6 

45.0 

.6 

44-4 

43-8 

.6 

43.2 

42.6 

■7 

S3- 9 

53-2 

52-5 

•7 

51.8 

511 

■7 

50-4 

49  7 

8 

61.6 

60.8 

60.0 

.8 

59.2 

58.4 

.8 

57-6 

568 

69.3     1      68.4     1      67.5 

6'^.  6 

6^7 

9     1     64.8     '•     63.9  _  I 

49 


10°. 

' 

L.  Sin.    1  d. 

L.  Tang-,     d.      L.  Cotg.  | 

L.  Cos.     d. 

0 

9. 26  967 

9.24  632 

0.75  368 

9.99  335 

60 

I 

9.24  089 

71 

9.24  706 

73 

0.75  294 

9 

99  333 

59 

2 

9.24  no 

9.24779 

0.75  221 

9 

99  33i 

58 

3 

9.24  181 

7' 
72 

9.24  853 

73 

0.75  i47 

9 

99  328 

3 

2 

57 

4 

9.24  253 

9.24  926 

74 

0.75  074 

9 

99  326 

56 

5 

9.24  324 

9.25  000 

0.75  000 

9 

99  324 

55 

G 

9.24  395 

7' 
71 

9.25  073 

73 

0.74  927 

9 

99  322 

3 

54 

7 

9.24  466 

9.25  i46 

73 

0.74  854 

9 

99319 

53 

8 

9.24  536 

9.25  219 

0.74  781 

9 

99317 

52 

9 

9.24  607 

71 

9.25  292 

0.74  708 

9 

99315 

5i 

10 

9.24  677 

71 

70 

9.25365 

72 
73 

0.74  635 

9 

99  3i3 

3 

50 

1 1 

9.24  748 

9.25437 

0.74  563 

9 

99  3 10 

49 

12 

9.24  818 

70 

9.25  5io 

72 

0.74  490 

9 

99  3o8 

48 

l3 

9.24  888 

9.25  582 

0.74418 

9 

99  3o6 

47 

70 

73 

2 

i4 

9.24  958 

70 

9.25  655 

72 

0.74345 

9 

99  3o4 

3 

46 

i5 

9.25  028 

9.25  727 

0.74  273 

9 

99  3oi 

45 

i6 

9.25  098 

70 

9.25799 

72 

0.74  201 

■9 

99299 

2 

44 

17 

9.25  168 

69 

9.25  871 

72 

0.74  129 

•9 

99297 

3 

43 

i8 

9.26  287 

9.25  943 

0.74  057 

9 

99  294 

42 

'9 

9.25  307 

69 

69 
6q 

9.26  015 

0.73  985 

9 

99  292 

4i 

20 

9.25  376 

9.26  086 

0.73  914 

9 

99  290 

40 

21 

9.25  445 

9.26  i58 

71 

0.73  842 

9 

99  288 

39 

22 

9.25  5i4 

69 

9.26  229 

0.73  771 

9- 

99  285 

38 

23 

9.25  583 

9.26  3oi 

72 

0.73  699 

9 

99  283 

37 

69 

71 

2 

24 

9.25  652 

69 
69 

9.26  372 

0.73  628 

9 

99  281 

36 

26 

9.25  721 

9.26443 

0.73  557 

9 

99278 

35 

26 

9.25  790 

9.26  5i4 

7' 

0.73486 

9 

99276 

34 

68 

71 

2 

27 

9.25  858 

69 
68 
68 

9.26  585 

0.73415 

9- 

99  274 

3 

33 

28 

9.25  927 

9.26  655 

0.73345 

9- 

99271 

32 

29 

9.25  995 

9.26  726 

7« 

0.73  274 

9- 

99  269 

3i 

30 

9.26  o63 

9.26  797 

0.73  2o3 

9 

99267 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

T 

' 

79°  30'. 

1 

PP 

.1 

74 

73 

7a 

.1 

71 

70 

69 

.1 

68 

3 

7-4 

7? 

7 

2 

7-« 

7.0 

6.9 

6.8 

0.3 

.2 

14.8 

14.6 

M 

4 

.2 

14.2 

14.0 

13.8 

.2 

.3-6 

0.6 

•3 

22.2 

21.9 

31 

6 

•3 

21.3 

21.0 

20.7 

•3 

20.4 

0.9 

•4 

39.6 

29.3 

28 

8 

•4 

28.4 

28.0 

27.6 

•4 

27.2 

1.2 

■5 

370 

36.5 

3b 

0 

•5 

35- S 

350 

345 

•5 

340 

'S 

44-4 

43-8 

43 

2 

.6 

42.6 

42.0 

4'-4 

.6 

40.8 

1.8 

•7 

5'.8 

St.i 

50 

4 

•7 

49-7 

49.0 

48.3 

•7 

47.6 

2.1 

.8 

59-2 

58.4 

S7 

6 

.8 

56.8 

56.0 

55-2 

.8 

54-4 

2.4 

.g    1    66.6    !    65.7 

64 

8 

63.0 

62.1 

^^ 

1^ 

61.2 

—2.^ 

5o 


10 

0 

30 

'. 

/ 

L.  Sin.       d. 

L.  Tang.     d.  |  L.  Cotg. 

L.  Cos.     d. 

30 

30 

9.26  o63    ': 

9.26  797 

0.73  203 

9-99  267 

70 

ii 

9.26  i3i       ^g 

9.26  867 

70 

0.73  i33 

9- 

gg  264 

2 

29 

•ii 

9.26  207 

9,26  937 

0.73  o63 

9- 

99  262 

28 

33 

9.27  008 

0.72  gg2 

9- 

99  260 

27 

68 

70 

3 

34 

9.26  335 

68 

9.27  078 

0.72  922 

9- 

99257 

2 

26 

35 

9.26  4o3 

9.27  i48 

0.72  852 

9- 

99255 

25 

36 

9.26  470 

67 

9.27  218 

70 

0.72  782 

9- 

gg  252 

24 

37 

9.26  538 

6-T 

9.27  288 

69 

0.72  712 

9- 

99  25o 

2 

23 

3« 

9.26  6o5    ;    ^ 

9.27  357 

0.72  643 

9- 

gg  248 

3 
2 

0 

22 

39 

9.26  672 

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67 

67 
67 
67 

9.27  427 

70 
69 

0.72  573 

9- 

gg  245 

21 

20 

40 

9.26  739 

g.27  4g6 

0.72  5o4 

9- 

gg  243 

4i 

9 .26  806 

g.27  566 

70 
69 
69 

0,72  434 

9- 

gg  241 

3 

19 

42 

9.26  873 

g.27  635 

0.72  365 

9- 

gg  238 

18 

43 

9.26  940 

9.27  704 

0,72  296 

9- 

gg  236 

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67 

69 

3 

44 

9.27  007 

66 

g.27  773 

69 

0.72  227 

9- 

gg  233 

2 

16 

4b 

9.27073 

9.27  842 

0.72  i58 

9- 

gg  23i 

i5 

46 

9.27  i4o 

67 
66 

9.27  911 

69 
69 
69 

0^72  089 

9- 

gg  22g 

3 

i4 

47 

9.27  206 

67 

g.27  gSo 

0.72  020 

9- 

gg  226 

2 

i3 

48 

9.27273 

66 
66 

66 
66 

9.28  049 

0.71  gSi 

9- 

gg  224 

12 

49 

9.27  33g 

9.28  117 

69 

68 
60 

0.71  883 

9- 

99  221 

2 

1 1 
10 

50 

9.27  405 

9.28  186 

0.71  8 1 4 

9- 

99219 

5i 

9.27471 

9,28  254 

0.71  746 

9- 

99  217 

3 

9 

52 

9.27  537 

9.28  323 

68 

0.71  677 

9- 

99  2l4 

8 

53 

9.27  602 

65 

9.28  391 

0.71  609 

9- 

99  212 

7 

66 

68 

3 

54 

g.27  668 

66 

9.28  459 

68 

0.71  541 

9- 

99  209 

2 

6 

56 

9,27  734 

65 

9.28  627 

68 

0.71  473 

9- 

gg  207 

5 

56 

9.27  799 

9.28  095 

0.71  4o5 

9- 

99  2o4 

4 

6=i 

67 

^7 

9.27  864 

66 

9.28  662 

68 

0.71  338 

9- 

gg  202 

2 

3 

58 

g.27  930 

65 

9.28  730 

68 

0.71  270 

9- 

gg  200 

2 

59 

9.27995 

9.28  798 

67 

0.71  202 

9- 

99  197 

I 

60 

9.28  060  I 

9.28  865 

0.71  135 

9- 

99  '95 

0 

L.  Cos.       d. 

L.  Cotg.  !   d.  1  L.  Tang. 

L.  Sin.     d. 

f 

79°.                                                       1 

PP 

.1 

70         69 

68 

.1 

67 

66 

.1 

65 

3 

7.0 

6-0 

6.8 

6.7 

6.6 

6.S 

0.3 

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14.0 

13-8 

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13.0 

0.6 

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21.0 

20.7 

20.4 

3 

20.1 

19.8 

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19.5 

0.9 

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28.0       27.6 

27.2 

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26.8 

26.4 

4 

26.0 

1.2 

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350           34- S 

340 

•5 

33-5 

330 

•5 

325 

1-5 

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42.0 

41.4 

40.8 

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40.2 

39-6 

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39.0 

1.8 

•7 

49.0 

48.3 

47.6 

•7 

46.9 

46.2 

•7 

45.S 

2.1 

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56.C 

55-2 

54-4 

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53-6 

528 

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52.0 

2-4 

6-5.0     1      62.1 

61.2 

.  g      !      60. 3 

-^L^ lUi 

5i 


11°. 

' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.28  oGo 

65 

6s 

9.28865 

68 
67 

0.71  185 

9-99  >95   ! 

60 

I 

9.28  125 

9.28  933 

0.71  067 

9 

99  192 

i 
2 

59 

2 

9.28  190 

64 
65 

9.29  000 

67 

67 

0.71  000 

9 

99  '90 

58 

3 

9.28  254 

9.29  067 

0.70  988 

9 

99  i«7 

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2 

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4 

9.28  819 

65 

9.29  i34 

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0.70  866 

9 

99  '85 

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56 

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9.28  384 

64 

9.29  201 

67 

0.70799 

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99  182 

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6 

9.28448 

9.29  268 

0.70  782 

9 

99  180 

54 

64 

67 

3 

7 

9.28  5l2 

65 

9.29  335 

67 
66 
67 

0.70  665 

9- 

99  177 

2 

53 

8 

9.28  577 

64 
64 

9.29  4o2 

0.70  598 

9- 

99  175 

52 

9 

9.28  64 1 

9.29  468 

0.70  532 

9 

99172 

DI 

10 

9.28  705 

64 
64 

9.29  535 

66 
67 

0.  70  465 

9 

99  '70 

50 

1 1 

9.28  769 

9.29  601 

0.70  399 

9- 

99  167 

49 

12 

9.28  833 

9.29  668 

0.70  832 

9- 

99  '65 

48 

i3 

9.28  896 

63 
64 

9.29  734 

66 
66 

0.70  266 

9- 

99  162 

i 

2 

47 

i4 

9.28  960 

64  - 

9.29  800 

66 

0.70  200 

9- 

99  160 

3 

46 

lb 

9.29  024 

9.29  866 

0.70  184 

9- 

99  '^7 

45 

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9.29  087 

63 
63 

9.29  982 

66 

0.70  068 

9- 

99  155 

3 

44 

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9.29  i5o 

64 
63 

63 

9.29998 

66 

0.70  002 

9- 

99  '52 

2 

43 

i8 

9.29  2l4 

9.80  064 

0.69  986 

9- 

99  '50 

42 

'9 

9.29  277 

9.80  i3o 

66 

0.69  870 

9 

99  '4? 

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4i 

20 

9.29  340 

9.30  195 

0.69  805 

9- 

99  145 

40 

63 

66 

2  1 

9.29  4o3 

63 

9.80  261 

65 

0.69  789 

9- 

99  142 

2 

39 

2  2 

9.29  466 

9.80  826 

0.69  674 

9- 

99  i4o 

88 

23 

9.29  529 

63 
62 

9.80  891 

t>5 
66 

0.69  609 

9- 

99  137 

2 

37 

24 

9.29  591 

63 

9.30  457 

65 

0.69  543 

9- 

99  135 

3 

86 

2b 

9.29  654 

9.80  522 

0.69  478 

9- 

99  '32 

85 

26 

9.29  716 

62 

9.80  587 

bS 

0.69  4i8 

9- 

99  i3o 

84 

27 

9.29  779 

63 
62 

9.80  652 

65 
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0.69  348 

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99  127 

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0.69  288 

9- 

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32 

29 

9.29  903 

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0.69  218 

9- 

99  122 

3i 

30 

9.29  966 

63 

9.80  846 

64 

0.69  1 54 

9- 

99  "9 

30 

L.  Cos.       d. 

L.  Cotg.      d.  '  L.  Tang.  | 

L.  Sin.      d. 

' 

78°  30 . 

PP 

I 

63 

67 

66 

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65 

64 

63 

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63 

3 

6.8 

6.7 

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6.5 

6.4 

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335 

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L. 

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2.7 

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L.  Sin.    [  d.  1 

L.  Tang,  j  d. 

L.  Cotg. 

L.  Cos.     d.| 

30 

9.29  966 

9.30  846 

6s 

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9.99  119 

30 

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9.30  91 1 

64 

0.69  089 

9 

99  117 

29 

32 

9.30  090 

61 

9 .3o  975 

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0.69  025 

9 

99  ii4 

28 

33 

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9.31  o4o 

0.68  960 

9 

99  112 

27 

62 

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9.30  2l3 

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0.68  896 

9 

99  "09 

26 

35 

9.30275 

61 

9.31  168 

0.68  832 

9 

99  106 

25 

36 

9.30  336 

9.31  233 

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9.31  297 

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9.31  36i 

0.68  689 

9 

99  099 

22 

39 

9.  3o  521 

9.31  425 

64 

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0.68  575 

9 

99  096 

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21 

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9.  3o  582 

61 

9.31  489 

63 
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0.68  5ii 

9 

99  098 

20 

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61 

9.3:  552 

0.68  448 

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9.30  704 

61 

9.3t  616 

63 

0.68  884 

9 

99  088 

18 

43 

9.30  765 

9.81  679 

0.68  821 

9 

99  086 

17 

61 

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3 

44 

9.30  826 

61 

9.81  748 

63 

0.68  257 

9 

99  o83 

16 

45 

9.30  887 

60 

9.81  806 

0.68  194 

9 

99  080 

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46 

9.30  947 

9.81  870 

64 

0.68  180 

9 

99078 

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61 

63 

3 

47 

9.31  008 

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9.31  988 

63 

0.68  067 

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48 

9.31  068 

61 

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9.31  129 

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63 

0.67  941 

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63 

3 

50 

9.31  189 

61 

9.82  122 

63 
63 

0.67  878 

9 

99  067 

10 

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9.31  250 

60 

9.82  i85 

0.67  815 

9 

99  064 

3 
2 

9 

52 

9,31  3io 

60 

60 

9.82  248 

0.67  752 

9 

99  062 

8 

53 

9.81  370 

9.82  3ii 

63 

62 

0.67  689 

9 

99  059 

3 

7 

54 

9.31  43o 

60 

9.82  878 

63 
62 

0.67  627 

9 

99  o56 

6 

55 
56 

9. 3 1  490 
9.31  549 

59 

9.82436 
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0.67  564 
0.67  5o2 

9 
9 

99  o54 
99  o5i 

3 

5 

4 

60 

63 

3 

57 

9.81  609 

fir. 

9.32  56i 

62 

0.67  489 

9 

99  o48 

2 

3 

58 

9.3i  669   i 

9.32  628 

0.67  877 

9 

99  o46- 

2 

59 

9.3i  728       59 

9.82685 

6a 

0.67  315 

9 

99  o48 

3 

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60 

9.31 788  1 

9.82  747 

0.67  258 

9 

99  o4o 

0 

L.  Cos.       d.      L.  Cotg.      d.   i  L.  Tang.  | 

L.  Sin.     d. 

' 

78°.                                                       1 

PP 

.1 

1    65 

64 

63 

.1 

62 

61 

60 

•  I 

59 

3 

6.5       6.4 

6.3 

6.2 

6.1 

6.0 

5-9 

0.3 

.2 

13.0      12.8 

12.6 

.2 

12.4 

12.2 

12.0 

.2 

II. 8 

0.6 

•3 

19-5 

19.2 

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•3 

18.6 

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18.0 

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17.7 

0.9 

4 

26.0 

25.6 

25.2 

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24.8 

24.4 

24.0 

4 

23.6 

1.2 

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32- S 

32.0 

3I-5 

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31.0 

30-5 

30.0 

■5 

29.5 

1-5 

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39-0 

3«.4 

37.8 

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37-2 

36.6 

36.0 

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35-4 

1.8 

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45- 5 

44.8 

44.1 

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43-4 

42.7 

42.0 

■7 

41.3 

2.1 

.8 

52.0 

51-2 

50.4 

.8 

49.6 

48.8 

48.0 

.8 

47.2 

2.4 

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53 


12°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d.  1  L.  Cotg. 

L.  Cos. 

d. 

0 

9.31  788 

59 
60 

9.32  747 

63 
62 

0.67  253 

9.99  o4o 

60 

I 

9.31  847 

9.32  810 

0.67  190 

9 

99  o38 

3 

59 

2 

9.3i  907 

59 
59 

9.32  872 

61 
62 

0.67  128 

9 

99  o35 

58 

3 

9.31  966 

9.32  933 

0.67  067 

9 

99  o32 

3 

57 

4 

9.32  025 

59 

9.32  996 

62 

0.67  005 

9 

99  o3o 

56 

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9.32  084 

9.33  o57 

62 
61 

0.66943 

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99027 

55 

6 

9.32  143 

59 
59 

9.33  119 

0.66881 

9 

99  024 

3 

54 

7 

9.32  202 

9.33  180 

62 

0.66  820 

9 

99  022 

3 

53 

» 

9.32  261 

58 

9.33  242 

61 

62 

0.66  758 

9 

99019 

52 

9 

g.32  319 

9.33  3o3 

0.66  697 

9 

99  016 

5i 

10 

9.32  378 

9.33  365 

0.66  635 

9 

99  oi3 

50 

1 1 

9.32  437 

58 

9.33  426 

61 

0.66  574 

9 

99  01 1 

3 

49 

12 

9.32  495 

58 

9.33487 

o.665i3 

9- 

99  008 

48 

1 3 

9.32  553 

9.33  548 

0.66  452 

9 

99  oo5 

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47 

59 

61 

1 

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9.32  612 

58 

9.33  609 

61 

0.66  391 

9- 

99  002 

46 

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9.32  670 

58 
58 
58 

9.33670 

0.66  33o 

9 

99  000 

45 

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9.32  728 

9.33  731 

6i 
61 
61 

0.66  269 

9- 

98997 

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44 

n 

9.32  786 

9.33  792 

0.66  208 

9- 

98  994 

3 

43 

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9.32  844 

58 

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58 

9.33  853 

0.66  147 

9- 

98991 

42 

19 

9.32  902 

9.33  9i3 

60 
6i 
60 
61 

0.66  087 

9- 

98  989 

3 

4i 

20 

9.32  960 

9.33  974 

0.66  026 

9- 

98  986 

40 

21 

9.33  018 

9.34034 

0.65  966 

9- 

98  983 

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39 

22 

9,33  075 

58 

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0.65  905 

9- 

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23 

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9.34  i55 

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9- 

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37 

57 

60 

3 

24 

9.33  190 

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9.34215 

61 

0.65785 

9- 

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36 

2b 

9.33248 

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9- 

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35 

26 

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37 
57 

9.34336 

60 

0. 65  664 

9- 

98  969 

3 

2 

34 

27 

9.33  362 

58 

9.34396 

60 

0.65  6o4 

9- 

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28 

9.33  420 

9.34456 

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9- 

98964 

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29 

9.33477 

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9.34516 

60 

0. 65  484 

9- 

98  961 

3 

3i 

30 

9.33534 

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9.34576 

0.65  424 

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98958 

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30 

L.  Cos. 

d. 

L.  Cotg.      d.     L.  Tang.  | 

L.  Sin.     d.| 

' 

77° 30.                                                  1 

PP 

.1 

63 

63 

61 

.1 

60 

59 

58 

.1 

57 

3 

6.3 

6.2 

6. 

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6.0 

5-9 
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5.8 

5-7 

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12.6 

12.4 

12. 

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2 

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3 

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0 

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L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.33  534 

57 

s6 

9.34576 

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60 

0.65  424 

9.98958 

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56 
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98  938 

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9.33  987 

9.35  o5i 

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0.64  949 

9- 

98  936 

22 

39 

9.34  043 

9.35  III 

59 

0.64889 

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98  933 

3 
3 

21 

40 

9 . 34  1 00 

57 
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98  930 

20 

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9.35  229 

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0.64  771 

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98  927 

19 

42 

9.34  212 

9.35  288 

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9- 

98  924 

18 

43 

9.34268 

56 
56 
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59 
58 
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0. 64  653 

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44 

9.34324 

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45 

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98  916 

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46 

9.34436 

56 
55 

9.35  523 

59 
58 

0.64477 

9 

98913 

3 

i4 

47 

9.34  491 

56 

9.35  58i 

59 

0.64  419 

9 

98  910 

3 

i3 

48 

9-34  547 

9.35  64o 

0.64  36o 

9 

98907 

12 

49 

9.34  602 

55 

9.35  698 

58 

0.64  3o2 

9 

98  904 

3 

1 1 

56 

59 

50 

9.34  658 

9.35  767 

58 
58 

0.64243 

9 

98  901 

3 

10 

5i 

9.34  713 

56 

9.35  815 

0.64  i85 

9 

98898 

2 

9 

52 

9.34  769 

9.35  873 

58 

0.64  127 

9 

98  896 

8 

53 

9.34  824 

55 

9.35  931 

0 .  64  069 

9 

98893 

7 

55 

58 

6 

64 

9.34  879 

9.35  989 

58 
58 

0.64  on 

9 

98  890 

3 

6 

55 

9.34934 

9.36  047 

0.63  953 

9 

98  887 

5 

56 

9.34  989 

55 

9.36  io5 

0.63  895 

9 

98884 

4 

57 

9.35  o44 

55 

9.36  i63 

5« 
58 

0. 63  837 

9 

98881 

i 
3 

3 

58 

9.35  099 

9.36  221 

0.63  779 

9 

98878 

3 

3 

2 

59 

9.35  i54 

55 
55 

9.36  279 

58 
57 

0.63  721 

9 

98  875 

I 

60 

9.35  209 

9.36  336 

0.63  664 

9 

98  872 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

77^ 

1 

PP 

.1 

60 

59 

58 

.1 

57 

56 

.1 

55 

3 

6.0 

5-9 

5.8 

5-7 

5.6 

5-5 

0-3 

.2 

12.0 

11.6 

.2 

II. 4 

II. 2 

.2 

II. 0 

0.6 

•3 

18.0 

17-7 

17.4 

•3 

17. 1 

16.8 

•3 

16.5 

0.9 

4 

24.0 

23.6 

23.2 

•4 

22.8 

22.4 

•4 

22.0 

1.2 

•5 

30.0 

29  5 

29.0 

•5 

28.5 

28.0 

•5 

275 

1-5 

.6 

36.0 

35-4 

34-8 

.6 

34-2 

33-6 

.b 

330 

1.8 

•  7 

42.0 

41-3 

40.6 

.7 

39-9 

39-2 

.7 

385 

2.1 

.8 

48.0 

47.2 

46.4 

.8 

45-6 

44.8 

.8 

44.0 

2.4 

•Q 

•9     1      51-3      1      50-4     1 

2-7 

55 


13^ 


L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.Cos.      d. 

0 

9.35  209 

9.36  336 

58 
58 

0. 63  664 

9.98  872 

60 

I 

9.35263 

9.36  394 

0.63  606 

9 

98869 

59 

2 

9.35  3i8 

9.36  452 

0. 63  548 

9 

98  867 

58 

3 

9.35  373 

55 
54 

9.36  509 

57 
57 

0.68  491 

9 

98864 

3 

3 

57 

4 

9.35  427 

9.36  566 

58 

0.68434 

9 

98  861 

3 

56 

5 

9.35481 

9.36  624 

0.68  376 

9 

98  858 

55 

6 

9.35  536 

55 
54 

9-36  681 

57 
57 

0.63  819 

9 

98855 

i 

3 

54 

7 

9.35  590 

9-36  788 

0.63  262 

9 

98  852 

53 

8 

9.35  644 

9-36  795 

0.63  205 

9 

98849 

52 

9 

9.35  698 

9-36  852 

57 

0.68  i48 

9 

98846 

5i 

10 

9 .  35  752 

54 
54 

9.36  909 

57 

0.63  091 

9 

98843 

3 

3 

50 

49 

1 1 

9.35  806 

9.36  966 

0.63  o34 

9 

98  84o 

12 

9.35  860 

9.37  028 

0.62  977 

9 

98  887 

48 

i3 

9.35  914 

54 

9.37  080 

57 

0.62  920 

9 

98834 

3 

47 

i4 

9.35  968 

54 

9.37187 

57 
56 

0.62868 

9 

98  83i 

46 

i5 

9-36  022 

9.87  Ig8 

0.62  807 

9 

98828 

45 

i6 

9.36  075 

53 

9.87  250 

57 
56 

0.62  75o 

9 

98825 

44 

17 

9.36  129 

54 

9.87  3o6 

0.62  694 

9 

98822 

3 

43 

i8 

9.36  182 

9.87  363 

0.62  687 

9 

98  819 

42 

19 

9.36  236 

54 
53 

9.87  419 

5b 
57 

0.62  58i 

9 

98816 

i 
3 

4i 

20 

g.36  289 

9.87  476 

0.62  524 

9 

98813 

40 

21 

9.36  342 

53 

9.37  582 

56 
56 

0.62468 

9 

98  810 

39 

22 

9.36  395 

9.87588 

0.62  4l2 

9 

98807 

38 

23 

9.36  449 

54 

9.37644 

5^ 

0.62  356 

9 

98804 

3 

37 

24 

9.36  5o2 

53 

9.87  700 

56 

0.62  800 

9 

98801 

3 
3 

36 

25 

9.36  555 

53 

9.87756 

56 

0.62  244 

9 

98798 

35 

26 

9.36608 

53 

9.87  812 

50 

0.62  188 

9 

98795 

3 

34 

27 

9.35  660 

52 

9.87868 

56 
56 

0,62  182 

9 

98792 

3 

33 

28 

9-36  7i3 

9.87  924 

0.62  076 

9 

98789 

32 

29 

9.36  766 

53 
53 

9.87  980 

5& 
55 

0.62  020 

9 

98  786 

3 

3i 
30 

30 

9.36  819 

9.88  o85 

0.61  965 

9 

98  788 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin.      d. 

' 

76°  3C 

V. 

1 

PP 

.1 

58 

57 

56 

.1 

55 

54 

53 

3 

s! 

5-7 

S.6 

>-5 

5J 

5-3 

•3 

.2 

11.6 

14 

II. 2 

.2 

I 

.0 

10.8 

.2 

lao 

.6 

•3 

17.4 

17.1 

16.8 

•3 

I 

5-5 

16.2 

•3 

159 

•9 

■4 

232 

22.8 

22.4 

•4 

2 

J.O 

21.6 

•4 

21.2 

1.3 

•5 

29.0 

:8.5 

28.0 

•5 

2 

?-5 

27.0 

■  5 

26.5 

'i 

.6 

34.8 

34-2 

33-6 

.6 

3 

5.0 

324 

.6 

3'-8 

1.8 

.7 

40.6 

39-9 

39.2 

.7 

3 

J-s 

37-8 

•7 

37- 1 

2.1 

.8 

46.4 

(5-b 

44.8 

.8 

4 

J.O 

43.2 

.8 

42.4 

2-4 

ILL^ 

).5          48.6      1 

56 


13°  30 . 


• 

L.  Sin. 

d. 

L.  Tang,  i  d. 

L.  Cotgf. 

L.  Cos.    !  d. 

30 

9-36  819 

9.38  o35 

S6 

0.61  965 

9.98  783 

30 

52 

3 

3i 

9.36871 

53 

9.38  091 

<;6 

0.61  909 

9 

98  780 

3 

29 

32 

9.  36  924 

9.38  147 

o.6i  853 

9 

98777 

28 

33 

9.  36  976 

52 
52 

9.38  202 

55 
55 

0.61  798 

9 

98  774 

3 
3 

27 

34 

9.37  028 

53 

9.38  257 

56 

0.61  743 

9 

98  771 

26 

35 

9.37  081 

9.38  3i3 

0.61  687 

9 

98  768 

25 

36 

9.37  i33 

52 
52 

9.38  368 

55 
55 

0.61  632 

9 

98  765 

3 
3 

24 

■il 

9.37  i85 

52 

9.38  423 

56 

0.61  577 

9 

98  762 

3 

23 

38 

9.37  237 

9-38  479 

0.61    521 

9 

98  759 

22 

39 

9.37  289 

52 

9.38  534 

55 

0.61  466 

9 

98  756 

3 

21 

40 

9.37  34i 

9.38  589 

55 

55 

0.61  4i  I 

9 

98  753 

20 

4i 

9.37  393 

52 

9.38  644 

0.61  356 

9 

98  750 

i 

'9 

42 

9.37445 

9.38  699 

0.61  3oi 

9 

98  746 

18 

43 

9.37497 

s« 

9.38  754 

54 

0.61  246 

9 

98743 

i 
3 

17 

44 

9.37  549 

9.38808 

55 

0.61  192 

9 

98  740 

16 

45 

9.37  600 

9.38  863 

0.61  137 

9 

98737 

i5 

46 

9.37652 

9.38918 

55 

o.6i  082 

9 

98734 

i 

j4 

47 

9. 37  703 

9.38  972 

54 

0.61  028 

9 

98  73i 

i3 

48 

9.37  755 

9. 39  027 

0.60  973 

9 

98728 

12 

49 

9.37  806 

5' 

52 

51 

9. 39  082 

55 
54 

0.60  918 

9 

98725 

J 
3 

II 
10 

50 

9.37  858 

9.39  I  36 

0.60864 

9 

98  722 

5i 

9.37909 

51 

9.39  190 

55 

0.60  8io 

9 

98  719 

4 

9 

52 

9  .37  960 

9.39  245 

0.60  755 

9 

98  7i5 

8 

53 

9.38  01 1 

5' 

9.39299 

54 

0.60  701 

9 

98  712 

3 

7 

54 

9.38  062 

51 

9.39  353 

54 

0.60  647 

9 

98  709 

6 

55 

9-38  ii3 

9.39  407 

0.60  593 

9 

98  706 

5 

56 

9.38  i64 

51 
51 

9.39  461 

54 

54 

0.60  539 

9 

98  7o3 

i 
3 

4 

^7 

9-38  2i5 

51 

9.39  5i5 

54 

0.60  485 

9 

98  700 

3 

58 

9.38  266 

9.39  569 

0.60  43i 

9 

98  697 

2 

59 

9.38  3i7 

5' 

9. 39  623 

54 

0.60  377 

9 

98  694 

I 

60 

9.38  368 

9.39677 

0.60  323 

9 

98  690 

0 

L.  Cos.    i  d. 

L.  Cotgr. 

d. 

L.  Tang. 

L.  Sin.    [  d. 

76°.                                                      1 

PP 

.1 

56 

55 

.    1 

.1 

53 

52 

51 

.1 

4 

3 

5.6 

■;.■; 

5- 

4 

5-3 

5.2 

5-1 

0.4 

°-| 

.2 

II. 3 

II. 0 

10. 

i 

.2 

10.6 

10.4 

10.2 

.2 

0.8 

0.6 

•3 

16.8 

16.5 

16. 

2 

•3 

15.9 

15.6 

15-3 

•3 

1.2 

0.9 

•4 

22.4 

22.0 

21. 

6 

■4 

21.2 

20.8 

20.4 

■4 

1.6 

1.2 

•5 

28.0 

27.S 

27. 

0 

■5 

26.5 

26.0 

25.5 

■5 

2.0 

1.5 

.6 

33.6 

33.0 

32. 

4 

.6 

31.8 

31.2 

30.6 

.6 

2.4 

1.8 

■7 

39-2 

.38.5 

37- 

8 

•7 

37-1 

36.4 

35-7 

.7 

2.8 

2.1 

.8 

44.8 

44.0 

43- 

2 

.8 

42.4 

4. .6 

40.8 

.8 

3.2 

2.4 

50.4    1    49.5    1    48. 

6 

•9    '    47.7    '    46.8 

57 


14°. 

"o" 

L.  Sin.^ 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

9.38  368 

50 
51 

9.39677 

0.60  323 

9.98  690 

60 

I 

9.38418 

9.39731 

54 
54 

0.60  269 

9 

98687 

3 

59 

2 

9.38469 

50 

9.39785 

0.60  2l5 

9 

98684 

58 

•6 

9.38  519 

9.39838 

53 

0.60  162 

9 

98681 

3 
3 

^7 

4 

9.38  570 

50 

9.39  892 

0.60  108 

9 

98678 

56 

b 

9.38  620 

9.39  945 

0.60  055 

9 

98675 

55 

6 

9.38  670 

9.39999 

54 

0.60  001 

9 

98671 

54 

7 

9.38  721 

9.40  062 

54 

0.59  948 

9 

98668 

53 

8 

9.38771 

9.40  106 

0.59  894 

9 

98665 

52 

9 

9.38  821 

9.40  169 

53 

0.59  84i 

9 

98662 

3 

5i 

10 

9.38  871 

50 

9.40  212 

54 

0.59  788 

9 

98  659 

3 

50 

II 

9.38  921 

9.40  266 

0.59  734 

9 

98  656 

49 

12 

9.38971 

9.40  319 

0.59  681 

9 

98  652 

48 

1 3 

9.39  021 

50 

9.40  372 

53 

0.59  628 

9 

98649 

i 

47 

i4 

9.39  071 

50 

9.40425 

S3 

0.59  575 

9 

98646 

i 

46 

lb 

9.39  121 

9.40478 

0.59  522 

9 

98643 

45 

i6 

9.39  170 

49 

9.40  53i 

53 

0.59  469 

9 

98640 

44 

I? 

9.39  220 

so 

9.40  584 

53 

0.59  4i6 

9 

98  636 

4 
3 

43 

i8 

9.39  270 

9.40  636 

52 

0.59364 

9 

98633 

42 

19 

9.39  319 

49 
50 

49 
49 

9.40  689 

53 
53 
53 
52 

0.59  3i  I 

9 

98  63o 

3 

4i 

20 

9.39  369 

9.40  742 

0.59258 

9 

98  627 

40 

21 

9.39  4i8 

9.40795 

0.59  2o5 

9 

98623 

39 

22 

9.39  467 

9.40  847 

0.59  i53 

9 

98  620 

38 

23 

9.39  617 

50 

9.40  900 

53 

0.59  100 

9- 

98  617 

■^1 

24 

9.39566 

9.40  962 

0.59  o48 

9 

98614 

36 

2D 

9.39  615 

9.41  005 

0.58  995 

9 

98  610 

35 

26 

9.39  664 

49 

9.41  057 

52 

0.58943 

9 

98  607 

34 

27 

9.39713 

49 

9.41  109 

52 

0.58891 

9- 

98  6o4 

33 

2S 

9.39  762 

9.41  161 

0. 58  839 

9- 

98  601 

32 

29 

9.39  811 

49 
49 

9.41  2l4 

53 
52 

0.58  786 

9- 

98  597 

3i 

30 

9.39  860 

9.41  266 

0.58  734 

9- 

98  594 

30 

L.  Cos. 

d. 

L.  Cotg.      d.  I  L.  Tang. 

L.  Sin.    Id.l 

' 

75°  30 .                                                  1 

PP 

.1 

54 

53 

53 

.1 

51 

50 

49 

■  I 

,       3    1 

H 

5  3 

5- 

2 

51 

5.0 

4.9 

■4 

•3 

.2 

10.8 

ia6 

10. 

4 

.2 

I0.2 

lO.O 

9.8 

.2 

.8 

.6 

•3 

16.3 

15-9 

15- 

6 

•3 

15-3 

150 

14.7 

•3 

1.3 

•9 

•4 

21.6 

21.3 

20. 

8 

•4 

20.4 

20.0 

19.6 

■4 

1.6 

1.2 

•5 

27.0 

26.5 

26. 

0 

•5 

255 

25.0 

245 

•5 

2.0 

1.5 

.6 

324 

31.8 

3»- 

2 

.6 

30.6 

30.0 

29.4 

.6 

2-4 

1.8 

•7 

37-8 

371 

36. 

4 

■7 

35-7 

35- 0 

34-3 

.7 

2.8 

2.1 

.8 

43-2 

424 

4'- 

6 

.8 

4a  8 

40.0 

39.2 

.8 

3-2 

2.4 

.9    1    48.6    1    47.7    I    46. 

8 

^^^ 

^ 

3.6                2.7       1 

58 


14 

°30 

', 

/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.39  860 

49 
49 

9.41  266 

52 

52 

0.58  734 

9.98  594 

3 
3 

30 

3i 

9.39909 

9.41  3i8 

0. 58  682 

9 

98  591 

29 

32 

9.39  958 

48 

9.41  370 

52 

o,58  63o 

9- 

98  588 

28 

33 

9.40  006 

49 

9.41  422 

0.58  578 

9 

98  584 

27 

34 

9.40  055 

48 

9.41  474 

52 

0.58  626 

9 

98  58i 

3 

26 

35 

9.40  io3 

9.41  526 

0. 58  474 

9 

98  578 

2  5 

36 

9.40  l52 

48 

9.41  578 

51 

52 

0.58  422 

9 

98574 

3 

3 

24 

3? 

9  .40  200 

9.41  629 

0.58  371 

9 

98  571 

23 

38 

9.40  249 

48 

9.41  681 

0.58  3i9 

9 

98  568 

22 

39 

9.40  297 

9.41  733 

51 

0.58  267 

9 

98565 

4 

3 

3 

21 

40 

9.40  346 

48 
48 

9-4i  784 

0.58  216 

9 

98  56i 

20 

4i 

9.40  394 

9.41  836 

0.58  i64 

9 

98  558 

'9 

42 

9.40  442 

48 

9.41  887 

0.58  ii3 

9 

98555 

18 

43 

9.40  490 

9.41  939 

0. 58  061 

9 

98  55i 

17 

48 

51 

3 

44 

9.40  538 

48 

9.41  990 

0.58  010 

9 

98  548 

3 

16 

45 

9.40  586 

48 

9.42  o4i 

0.57  959 

9 

98545 

i5 

46 

9.40  634 

9.42  093 

0.57  907 

9 

98  54i 

i4 

48 

51 

3 

47 

9.40  682 

48 

9.42  i44 

0.57  856 

9 

98  538 

3 

i3 

48 

9.40  73o 

9.42  195 

0.57  805 

9 

98  535 

12 

49 

9.40  778 

47 
48 
48 

9.42  246 

51 
51 

0.57  754 

9 

98  53i 

3 

3 
4 

1 1 

50 

9.40  825 

9.42  297 

0.57  703 

9 

98  528 

10 

5i 

9.40  873 

9.42  348 

5' 

0.57  652 

9 

98  525 

9 

52 

9.40  921 

9.42  399 

0.57  601 

9 

98  521 

3 

8 

53 

9.40  968 

9.42  45o 

5' 

0.57  550 

9 

98  5i8 

7 

51 

3 

54 

9.41  016 

9.42  5oi 

SI 

0.57  499 

9 

98515 

4 

6 

55 

9.41  o63 

48 

9.42  552 

0.57448 

9 

.98  5ii 

5 

56 

9.41  III 

9.42  6o3 

51 

0.57  397 

9 

98  5o8 

4 

47 

50 

i 

57 

9.41  i58 

47 

9.42653 

51 

0.57  347 

9 

.98  505 

4 

3 

58 

9.41  205 

9.42  704 

0.57  296 

9 

.98  5oi 

2 

59 

9.4t  252 

47 
48 

9.42  755 

51 

0.57  245 

9 

.98498 

I 

60 

9.41  3oo 

9.42  8o5 

50 

0.57  195 

9 

.98494 

0 

L.  Cos. 

Ti 

L.  Cotg. 

d. 

1  L.  Tang. 

L.  Sin. 

d. 

/ 

75^                                                      1 

PP 

5a 

51 

50 

49 

48 

47 

4 

3 

.1 

52 

51 

50 

.1 

4.9 

4.8 

4-7 

.1 

0.4 

0.3 

.2 

10.4 

10.2 

10.0 

.2 

9.8 

q.b 

9.4 

.2 

0.8 

0.6 

•3 

IS.6 

15-3 

150 

3 

14.7 

14.4 

14.1 

•3 

12 

0.9 

•4 

20.8 

20.4 

20.0 

•4 

■   19.6 

19.2 

18.8 

•4 

1.6 

1.2 

•5 

26.0 

25.5 

25.0 

•s 

24- 5 

24.0 

23- 5 

•5 

2.0 

I  5 

.6 

31.2 

30.6 

30.0 

.6 

29.4 

28.8 

28.2 

.6 

24 

1.8 

•  7 

36-4 

35-7 

350 

•7 

34-3 

33.6 

329 

■7 

2.8 

2.1 

.8 

41.6 

40.8 

40.0 

.8 

39-2 

3«-4 

37  6 

.8 

3-2 

2.4 

.g   1     46.8     1     45  9     1     450      1         -9   '     44-1     ' 

43.2     i     42.3 

■9  1       3-6       1 

59 


15^ 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.    1  d. 

0 

9.41  3oo 

47 
47 

9.42  8o5 

51 

5° 

0.57  195 

9-98494 

3 
3 

60 

I 

9.41  347 

9.42  »56 

0.57  i44 

9.98  491 

59 

2 

9,41  394 

9.42  906 

0.57  094 

9.98  488 

58 

3 

9.41  44 1 

47 

9.42  957 

50 

0.57  043 

9.98484 

57 

4 

9.41  488 

9.43  007 

50 

0.56  993 

9.98481 

56 

5 

9-41  535 

9.43057 

0.56943 

9.98477 

55 

6 

9.41  582 

46 

9.43  108 

50 

0.56  892 

9.98474 

54 

7 

9.41  628 

9.43  i58 

50 

0. 56  842 

9.98471 

53 

8 

9.41  675 

9.43208 

0.56  792 

9.98  467 

52 

9 

9.41  722 

47 

9.43  258 

50 

0.56  742 

9.98464 

5i 

10 

9.4r  768 

9,43  3o8 

0.56  692 

9.98  460 

50 

1 1 

9.41  815 

47 
46 

9.43358 

50 

0.56  642 

9.98  457 

49 

12 

9.41  861 

9.43408 

0.56  592 

9.98  453 

48 

i3 

9.41  908 

47 
46 

9.43458 

5° 
■to 

0.56  542 

9.98  45o 

47 

i4 

9.4r  954 

9.43508 

0.56  492 

9.98447 

46 

i5 

9.42  001 

9.43558 

0.56  442 

9.98443 

45 

i6 

9.42  047 

46 
46 

9.43  607 

49 

0.56393 

■9.98440 

44 

17 

9.42  093 

9-43  657 

50 

0. 56  343 

9.98436 

43 

i8 

9.42  i4o 

9.43  707 

0.56  293 

9.98433 

42 

19 

9.42  186 

9.43756 

49 

0.56  244 

9.98  429 

4i 

20 

9.42  232 

46 
46 

9.43  806 

50 

49 
50 

0.56  194 

9.98  426 

40 

39 

21 

9.42  278 

9.43  855 

0.56  145 

9.98  422 

22 

9.42   324 

9.43905 

0.56  095 

9.98  419 

38 

23 

9.43   370 

46 
46 

9.43954 

49 
5° 

0.56  o46 

9.98415 

37 

24 

9.42  4i6 

9.44  oo4 

0.55  996 

9.98  4l2 

36 

25 

9.42  46i 

45 

9.44  o53 

0.55  947 

9.98  409 

35 

26 

9.42  507 

46 

9.44  102 

0.55898 

9.98  4o5 

34 

27 

9.42553 

46 

9.44  i5i 

50 

0.55  849 

9.98  4o2 

33 

28 

9.42  599 

9,44  201 

0.55  799 

9.98398 

32 

29 

9.42  644 

45 
46 

9.44  250 

49 
49 

0.55  750 

9.98395 

3i 

30 

9.42  690 

9.44  299 

0.55  701 

9.98  391 

30 

L.  Cos. 

IT 

L.  Cotg.  1  d.  1  L.Tang. 

L.Sin. 

d. 

74°  30 .                                                   1 

PP 

.1 

51 

50 

49 

48 

47 

46 

.1 

45 

4 

3 

5' 

5.0 

49 

.1 

4.8 

47 

4.6 

4-5 

0.4 

0.3 

.2 

10.2 

10. 0 

9.8 

.« 

9.6 

9.4 

9.2 

.2 

9.0 

0.8 

0.6 

•3 

iS-3 

150 

M-7 

•3 

14.4 

14. 1 

13.8 

•3 

135 

1.2 

0.9 

•4 

20.4 

20.0 

19.6 

•4 

19.2 

18.8 

.8.4 

•4 

18.0 

1.6 

1.2 

•5 

25- 5 

25.0 

=4-5 

■  5 

24.0 

23-5 

13.0 

•5 

22.5 

2.0 

« 5 

.6 

30.6 

300 

39.4 

.6 

28.8 

28.2 

27.6 

.6 

27.0 

2-4 

1.8 

:I 

35  Z 

350- 

34-3 

•7 

33-6 

329 

3?- 2 

•7 

31-5 

2.8 

2.1 

40.8 

40.0 

39-2 

.8 

384 

37- «> 

36.8 

.8 

36.0 

3-2 

2.4 

44' 

40.5   1      3.6    !       2.7       1 

60 


15°  30 


L.  Sin. 

d. 

L.  Tang.     d.      L.  Cotg. 

L.  Cos.    I  d. 

30 

9.42  690 

9.44  299 

49 
49 

0.55  701 

9.9b  391 

30 

3i 

9.42  735 

46 

9.44348 

0.55  652 

9 

98  388 

4 

29 

32 

9.42  781 

9.44  397 

0.55  6o3 

9 

98  384 

28 

33 

9.42  826 

46 

9-44  446 

49 

0.55554 

9 

98  38i 

3 
4 

27 

34 

9.42  872 

45 

9.44495 

49 

0.55  5o5 

9 

98377 

4 

26 

3b 

9.42  917 

9-44  544 

48 

0.55  456 

9 

98373 

2D 

36 

9.42  962 

45 

9.44  592 

0.55408 

9 

98  870 

i 

24 

46 

49 

4 

^7 

9.43  008 

9.44  64i 

0.55  359 

9 

98  366 

23 

38 

9.43  o53 

9.44  690 

48 
49 

49 
48 

0,55  3io 

9 

98  363 

22 

39 

9.43  098 

45 
45 

45 

9-44  738 

0.55  262 

9 

98  359 

4 

3 

4 

21 
19 

40 

9.43  i43 

9-44  787 

0.55  2l3 

9 

98  356 

4i 

9.43  188 

9.44  836 

0.55  164 

9 

98  352 

42 

9.43233 

9.44  884 

0.55  116 

9 

98349 

18 

43 

9.43  278 

45 

9.44933 

49 

48 
48 

0.55  067 

9 

98345 

4 

17 

44 

9.43  323 

43 

9.44  981 

0.55  019 

9 

98  342 

16 

4i) 

9.43367 

9.45  029 

0.54  971 

9 

98  338 

i5 

46 

9.43412 

45 
45 

9.45  078 

49 
48 

0.54  922 

9 

98334 

4 
3 

i4 

47 

9.43457 

9.45  126 

48 

0.54  874 

9 

98  33i 

i3 

4« 

9.43  502 

9-45  174 

0.54826 

9 

98  327 

12 

49 

9.43546 

44 

45 

9.45  222 

48 
49 

0.54778 

9 

98  324 

3 
4 

II 

50 

9.43  591 

44 

9.45  271 

48 
48 

0.54  729 

9 

98  320 

3 

10 

9 

5i 

9.43  635 

9.45  319 

0.54681 

9 

98  3i7 

52 

9.43680 

9.45  367 

0.54  633 

9 

98  3i3 

8 

53 

9.43  724 

44 

9.45  415 

48 
48 
48 
48 

0.54  585 

9 

98  309 

4 

7 

54 

9.43  769 

45 

9.45  463 

0.54537 

9 

98  3o6 

3 

6 

55 

9.43813 

44 

9.45  5i  I 

0.54489 

9 

98  302 

5 

56 

9.43  85? 

44 

9.45  559 

0.54441 

9 

98  299 

3 

4 

57 

9.43  901 

44 

9.45  606 

47 
48 

0.54  394 

9 

98  295 

4 

3 

58 

9.43946 

9.45554 

0.54346 

9 

98  291 

2 

59 

9.43  990 

44 
44 

9.45  702 

48 

48 

0.54  298 

9 

98288 

3 
4 

I 
0 

60 

9.44034 

9.45  750 

0.54  25o 

9 

98284 

L.  Cos. 

d. 

L.Cotg.  i  d. 

L.  Tang. 

L.  Sin. 

"dT 

' 

74°.                                                      1 

PP 

.1 

49 

48 

47 

.1 

46 

45 

44 

.1 

4 

3 

4.9 

-    4-8 

4' 

7 

4.6 

45 

4.4 

0.4 

03 

.2 

9.8 

9.6 

9-' 

^ 

.2 

9-^ 

9.0 

8.8 

.2 

0.8 

0.6 

■3 

14.7 

14.4 

14. 

t 

•3 

13-8 

13-5 

13.2 

■3 

1.2 

0.9 

•4 

19.6 

19.2 

18. 

3 

■4 

18.4 

18.0 

17.6 

•4 

1.6 

1.2 

•5 

245 

24.0 

23- 

> 

•s 

23.0 

22.5 

22.0 

•s 

2.0 

i-S 

.6 

29.4 

28.8 

28. 

2 

.6 

27.6 

27.0 

26.4 

.6 

2.4 

1.8 

•7 

34-3 

33-6 

32- 

? 

•7 

32.2 

3I-S 

30.8 

•7 

2.8 

2.1 

.8 

39  2 

3»-4 

37- 

D 

.8 

36.8 

36.0 

35-2 

.8 

3-2 

2.4 

■  g  1    44-1     1    43-2    1    42- 

L. 

.9  1    41.4    1    40.5 

_^^^ 

61 


16°. 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.    i  d. 

0 

9.44  o34 

9.45  750 

0.54  25o 

9.98  284 

60 

1 

9.44  078 

9.45  797 

48 

0.54  2o3 

9 

.98  281 

59 

2 

9.44   122 

9.45  845 

0.54  i55 

9 

.98277 

58 

3 

9.44   166 

44 

9.45892 

47 
48 

0.54  108 

9 

.98  273 

57 

4 

9.44  210 

9.45  940 

47 

0.54  060 

9 

.98  270 

56 

5 

9.44253 

9.45  987 

0.54  oi3 

9 

.98  266 

55 

6 

9.44  297 

44 
44 

9.46035 

48 
47 

0.53965 

9 

.98  262 

54 

7 

9.44341 

9.46  082 

48 

0.53  918 

9 

.98  259 

53 

8 

9-44  385 

9-46  i3o 

0.53  870 

9 

.98255 

52 

9 

9.44428 

43 

9.46  177 

47 

0.53  823 

9 

.98  25l 

5i 

10 

9.44  472 

44 

9.46  224 

47 
47 

0.53  776 

9 

98  248 

4 

60 

II 

9.44  5i6 

9.46  271 

48 

0.53  729 

9 

98244 

49 

12 

9.44559 

9.46  319 

0.53  681 

9 

.98  24o 

48 

i3 

9.44  602 

43 

9-46  366 

47 

0.53  634 

9 

98  237 

47 

i4 

9.44  646 

9.464i3 

47 

0.53  587 

9 

.98  233 

46 

i5 

9.44689 

9.46  460 

0.53  540 

9 

.98  229 

45 

i6 

9.44733 

44 

9.46  507 

47 

0.53  493 

9 

.98  226 

44 

'7 

9.44  776 

9.46  554 

47 

0.53446 

9 

.98  222 

A 

43 

i8 

9.44  819 

9.46  601 

0.53  399 

9 

98  218 

42 

'9 

9.44862 

43 
43 

43 

9-46  648 

47 
46 

47 

0.53  352 

9 

.98215 

4i 

20 

9.44  905 

9.46  694 

0.53  3o6 

9 

.98  211 

40 

39 

21 

9.44948 

9.46  74i 

0.53  259 

9 

98  207 

22 

9.44  992 

9.46  788 

47 

0.53  212 

9 

98  2o4 

38 

23 

9.45  035 

43 

9.46835 

47 

0.53  i65 

9 

98  200 

37 

24 

9.45077 

42 

9.46  88 1 

46 

0.53  1 19 

9 

98   196 

36 

25 

9.45  120 

43 

9.46  928 

47 

0.53  072 

9 

98   192 

35 

26 

9.45  i63 

43 

9.46975 

47 

0.53  025 

9 

98   189 

34 

27 

9.45  206 

43 

9.47  021 

46 

0.52  979 

9 

98   185 

33 

28 

9.45  249 

9.47  068 

47 

0.52  932 

9 

98   181 

32 

29 

9.45  292 

43 
42 

9.47  ii4 

40 

46 

0.52  886 

9 

98   177 

3 

3i 
30 

30 

9.45  334 

9.47  160 

0.52  84o 

9 

98   174 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.SIn.    id.j 

' 

73 

°30 

1 

PP 

I 

48 

47 

46 

45 

44 

43 

.1 

4a 

4 

3 

4.8 

4-7 

4.6 

.1 

4-S 

4-4 

4-3 

4.2 

0.4 

0.3 

.2 

9.6 

9-4 

^o 

.2 

9.0 

8.8 

8.6 

.2 

8.4 

0.8 

0.6 

•3 

M-4 

.14  1 

13-8 

•3 

'3-5 

13.2 

12.9 

•3 

12.6 

1.2 

0.9 

•4 

19.2 

18.8 

18.4 

•4 

18.0 

17.6 

17.2 

•4 

16.8 

1.6 

1.3 

5 

24.0 

23- S 

23.0 

•5 

22.5 

22.0 

21-5 

■5 

21.0 

2.0 

1.5 

.6 

38.8 

28.2 

27.6 

.6 

27.0 

26.4 

25.8 

.6 

25-2 

2.4 

1.8 

•7 

3^6 

329 

322 

.7 

31S 

30.8 

30.1 

•7 

29.4 

3.8 

3.1 

.8 

38.4 

37-6 

36.8 

.8 

36.0 

35  2 

3  ••4 

.8 

33-6 

3-2 

2.4 

.X^ 

3^ 

38.Z. 

?7-«      i" 

62 


le 

°30 

'• 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.45344 

9.47  i6o 

0.52  84o 

9.98  174 

30 

3i 

9.45  377 

43 

9.47  207 

47 
46 

0. 52  793 

9 

.98  170 

4 

29 

32 

9.45  419 

9.47  253 

0.52  747 

9 

.98  166 

28 

33 

9.45  462 

43 

9.47299 

46 

0.52  701 

9 

.98  162 

4 

27 

34 

9.45504 

42 

9.47346 

47 
46 

0.52  654 

9 

.98  1 59 

3 

26 

35 

9-45  547 

9.47  392 

0.52  608 

9 

.98  155 

25 

36 

9.45  589 

9-47  438 

46 
46 
46 

0.52  562 

9 

.98  i5i 

i 

24 

37 

9.45  632 

42 

9-47  484 

0.52  5i6 

9 

.98  i47 

23 

38 

9.45674 

9.47  53o 

0.62  470 

9 

98  i44 

22 

39 

9.45  716 

42 

9.47  576 

46 
46 

0.52  424 

9 

.98  i4o 

4 

21 

4 

40 

9.45  758 

9.47  622 

46 
46 

0.52  378 

9 

98  i36 

20 

4i 

9.45  801 

42 

9.47668 

0.52  332 

9 

.98  l32 

4 

19 

42 

9.45  843 

42 

9.47  714 

46 

0.52  286 

9 

.98  129 

18 

43 

9.45885 

9.47  760 

0.52  240 

9 

.98  125 

4 

17 

42 

46 

4 

44 

9.45  927 

9.47  806 

46 

0.52  194 

9 

.98  121 

4 

16 

4b 

9.45  969 

9.47  852 

0.52  i48 

9 

.98  117 

i5 

46 

9.46  01 1 

42 

9.47897 

45 
46 

0.52  io3 

9 

.98  ii3 

4 

3 

i4 

47 

9.46  o53 

9-47  943 

46 

0.52  067 

9 

.98  1 10 

4 

i3 

48 

9.46  095 

9.47989 

0.62  01 1 

9 

.98  106 

12 

49 

9.46  i36 

9.48  035 

46 

0. 5i  965 

9 

.98  102 

4 

1 1 

42 

45 

4 

50 

9.46  178 

9.48080 

46 

o.5i  920 

9 

.98  098 

10 

bi 

9.46  220 

42 

9.48  126 

o.5i  874 

9 

.98  094 

9 

62 

9  .46  262 

9.48  171 

46 

o.5i  829 

9 

.98  090 

8 

53 

9.46  3o3 

9.48  217 

o.5i  783 

9 

.98087 

i 

7 

42 

45 

4 

b4 

9.46  345 

41 

9.48  262 

45 
46 

o.5i  738 

9 

.98  o83 

4 

6 

55 

9.46  386 

9.48  307 

o.5i  693 

9 

.98  079 

5 

56 

9.46  428 

9.48  353 

o.5i  647 

9 

.98  075 

4 

4 

5? 

9.46  469 

9.48  398 

45 
45 

0.5 1  602 

9 

.98  071 

3 

58 

9.46  5i  I 

9.48443 

o.5i  557 

9 

.98  067 

2 

59 

9.46  552 

41 

9.48489 

46 

o.5i  5i  I 

9 

.98  o63 

4 

I 

42 

45 

3 

60 

9.46  594 

9.48  534 

o,5i  466 

9 

.98  060 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

73°.                                                       1 

PP 

47 

46 

45 

.1 

44 

43 

43 

.1 

41 

4 

3 

.1 

4-7 

4.6 

4-5 

4-4 

4-3 

4-2 

4.1 

0.4 

0.3 

.2 

9-4 

9.2 

9.0 

.2 

8.8 

8.6 

8.4 

2 

8.2 

0.8 

0.6 

3 

14.1 

'3.a 

I3-S 

.3 

13-2 

12.9 

12.6 

•3 

12.3 

1.2 

0.9 

•4 

18.8 

.8.4 

18.0 

•4 

.7.6 

17,2 

16.8 

•4 

.6.4 

1.6 

1.2 

■5 

23-5 

23.0 

22.5 

5 

22.0 

21. s 

21.0 

•5 

20.5 

2.0 

i-S 

.6 

28.2 

27.6 

27.0 

.6 

26.4 

25.8 

25.2 

.6 

24.6 

2.4 

1.8 

•7 

32-9 

32.2 

31-S 

■7 

30.8 

30.1 

29.4 

■7 

28.7 

2.8 

2.1 

.8 

37-6 

36.8 

36.0 

.8 

35.2 

34-4 

33.6 

.8 

32.8 

3-2 

2.4 

42.3  1    41.4 

.^7,1. 

36.9  1   3.6    1    2.7     1 

63 


L.  Sin.    !  d. 


17°. 

Tang.     d.  |  L.  Cotg. 


L.  Cos.     d. 


:;.46  594 


10 


i3 

i4 
i5 
16 

17 
18 

12. 
20 


21 
22 

23 

24 

25 

26 

27 

28 

29 

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46  635 
46676 
46  717 

46758 
46  800 
46  84i 

46  882 
46  923 
46964 


47  00; 


47  045 
47086 
47  127 

47  i63 
47  209 
47  249 

47  290 
47  33o 

47  371 


47  4ii 


47452 
47  492 
47  533 

47573 
47  6i3 
47654 

47  694 
47  734 

47  774 


9.47814 


48  534 


48  579 
48624 
48669 

48714 
48759 
48  8o4 

48  849 
48894 
48  939 


48984 


49  029 
49  073 
49  118 

49  i63 

49  207 

49  252 
49  296 

49  34i 
49  385 


49  43o 


49  474 
49  519 
49  563 

49  607 
49  652 
49  696 

49  740 

49784 
49  828 


49  872 


45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 

44 
45 
44 
44 
45 
44 
44 
44 
44 
44 


o.5i  466 


9 .  98  060 


o.5i  421 
o.5i  376 
o.5i  33i 

o.5i  286 
o.5i  241 
0.5 1  196 

o.5i  i5i 
0.5 1  106 
o.5i  061 


o.5i  016 


o.5o  971 
o.5o  927 
0.50882 

o.5o  837 
o.5o  793 
o.5o  748 

o.5o  704 
o.5o  659 
o.5o  615 


o.5o  570 


o.5o  526 
o.5o  481 
o.5o  437 

o.5o  393 
o,5o  348 
o.5o  3o4 

o.5o  260 
o.5o  216 
o.5o  172 


o.5o  128 


98  o56 
98  o52 

98048 
98044 

98  o4o 

98036 

98  o32 
98  029 
98  025 


98  017 
98  oi3 
98  009 

98  oo5 
98  oot 
97  997 

97993 
97989 
97  986 


97982 


97978 

97  974 
97970 

97  966 
97  962 
97958 

97954 

97950 
97  946 


9.97942 


L.  Cos.  I  d. 


L.  Cotg.  I  d.  L.  Tang. 


L.  Sin.   d. 


72°  30 . 


pp 

45 

44 

43 

.1 

.2 

43 

4« 

40 

.1 
.2 

4 

3 

.1 

.2 

45 
9.0 

Vs 

tl 

t: 

41 
8.2 

t.l 

0.4 
0.8 

0-3 
a6 

•3 

'3-5 

13.2 

i2.g 

•3 

12.6 

12-3 

12.0 

■3 

1.2 

0.9 

•4 
■5 
.6 

18.0 

22-5 
27.0 

.7.6 

22.0 
26.4 

17.2 

21.5 

25.8 

•4 
•5 
.6 

16.8 
21.0 
25.2 

16.4 
20.5 
24.6 

16.0 
20.0 
24.0 

•4 
•5 
.6 

1.6 
2.0 

2.4 

X.2 

1-5 
1.8 

•7 
.8 

31-5 
36.0 

30.8 

35-2 
39-6 

30.1 

344 

_^i7 

:l 

29.4 

33-6 

28.7 

28.0 
32.0 

•7 
.8 

2.8 
^1 

3.1 

2.4 

64 


17 

°30. 

/ 

L.  Sin. 

d. 

L.  Tang.  |  d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.47  8i4 

40 

9.49  872 

44 

0.5o   I2» 

9.97  942 

30 

3i 

9.47854 

40 

9.49  916 

44 

o.5o  084 

9 

97  938 

29 

32 

9.47  894 

9.49  960 

o.5o  o4o 

9 

97  934 

28 

33 

9.47  934 

40 

9 .  5o  oo4 

44 

0.49  996 

9 

97930 

27 

34 

9.47974 

9.50  o48 

44 

0.49  952 

9 

97  926 

26 

35 

9.48  oi4 

9.5©  092 

0.49  908 

9 

97922 

25 

36 

9.48054 

40 
40 

9.50  i36 

44 
44 

0.4986.4 

9 

97  9'8 

24 

■^7 

9.48  094 

39 

9.50  180 

0.49  820 

9 

97914 

23 

38 

9.48  i33 

9.50  223 

0.49777 

9 

97910 

22 

39 

9.48  173 

40 

9.5o  267 

44 

0.49  733 

9 

97  906 

21 

40 

9.48  2l3 

40 

9.5o  3i I 

44 

0.49  689 

9 

97902 

20 

4i 

9.48  252 

39 

9.50  355 

44 

0.49  645 

9 

97898 

'9 

42 

9.48  292 

9.50  398 

0.49  602 

9 

97  894 

18 

43 

9.48  332 

40 
39 

9.50  442 

43 

0.49  558 

9 

97  890 

'7 

44 

9.48  371 

9.5o485 

44 

0.49  515 

9 

97  886 

16 

45 

9.484II 

9.50  529 

0.49  471 

9 

97  882 

i5 

46 

9.48450 

39 
40 

9.50  572 

43 
44 

0.49  428 

9 

97878 

i4 

47 

9.48  490 

9.50  616 

43 

0.49  384 

9 

97874 

i3 

4« 

9.48  52g 

9.50  659 

0.49  341 

9 

97870 

12 

49 

9.48  568 

39 

9.5o  703 

44 

0.49  297 

9 

97  866 

1 1 

50 

g.48  607 

9.50  746 

0.49  254 

9 

97  861 

10 

bi 

9.48647 

9.5o  789 

44 

0.49  211 

9 

97837 

9 

52 

9.48686 

9.50  833 

0.49  167 

9 

97  853 

8 

53 

9.48  725 

39 
39 

9.50  876 

43 
43 

0.49  124 

9 

97  849 

7 

54 

9.48  764 

39 

g.5o  919 

43 

0.49  081 

9 

97845 

6 

55 

9.48803 

9.50  962 

0.49  o38 

9 

97  84i 

5 

56 

9.48842 

39 

39 

9.61  oo5 

43 
43 

0.48  995 

9 

97837 

4 

^7 

9.48  881 

39 

9.5i  o48 

0,48  952 

9 

97833 

3 

58 

9.48  920 

9. 5 1  092 

0.48  908 

9 

97  829 

2 

59 

9.48  959 

39 

9.51  135 

43 

0. 48  865 

9 

97825 

I 

39 

43 

60 

9.48998 

9.5i  178 

0.48  822 

9 

97  821 

0 

L.  Cos.       d. 

L.  Cotg. 

d.  !  L.  Tang. 

L.  Sin.      d. 

72°.                                                         1 

PP 

I 

44 

43 

4a 

.1 

4' 

40 

39 

.1 

5 

0-5 

4 

0.4 

4.4 

4-3 

4.2 

4.1 

4.0 

3  9 

.2 

8.8 

8.6 

8.4 

.2 

8.2 

8.0 

7.8 

.2 

I.O 

0.8 

•3 

13.2 

12.9 

12.6 

•3 

12.3 

I2.0 

II. 7 

•3 

i-S 

1.2 

•4 

,7.6 

17.2 

16.8 

•4 

.6.4 

16.0 

15-6 

•4 

2.0 

1.6 

•5 

22.0 

2I-5 

21.0 

•5 

20.5 

20.0 

'95 

•5 

2-5 

2.0 

.6 

26.4 

25.8 

25.2 

.6 

24.6 

24.0 

234 

.6 

3.0 

2.4 

•7 

30.8 

30.1 

29.4 

.7 

28.7 

28.0 

273 

•7 

3-5 

2.8 

.8 

35  2 

34-4 

33-6 

.8 

328 

32.0 

31.2 

.8 

4.0 

3-2 

9  1     39.6    i     38.7    1     37.8 

_r 

36.0     1       35.1 

—2. 

^^ iLJ 

65 


18^ 


L.Sin.    1  d.  1 

L.  Tangr. 

d. 

L.  Cotg. 

L.  Cos.      d. 

0 

9.48998 

39 
39 
39 
38 
39 
39 
38 
39 
39 
38 

39 
38 
38 

9. 5 1  178 

43 
43 
42 

0.48822 

9.97  821 

4 
S 

60 

I 

2 

3 

9 
9 
9 

49  037 
49  076 

49 115 

9.5i  221 
9. 5 1  264 
9.51  3o6 

0.48  779 
0.48736 
0.48694 

9 
9 
9 

97817 
97  812 
97808 

59 

58 
57 

4 
5 
6 

9 
9 
9 

49  i53 
49  192 

49  23l 

9.51  349 
9.5i  392 
9-5i435 

43 
43 
43 
42 
43 
43 

42 
43 
43 

0.48  65I 
0.48  608 
0. 48  565 

9 
9 
9 

97  8o4 
97  800 
97796 

56 
55 
54 

7 
8 

9 

9 
9 
9 

49  269 
49  3o8 
49347 

9.51478 
9.5i  520 
9.51  563 

0.48  522 

0.48480 
0  48437 

9 
9 
9 

97792 
97788 
97784 

53 

52 

5i 

10 

9 

49  385 

9. 5 1  606 

0.48  394 

9 

97  779 

50 

II 

12 

i3 

9 
9 
9 

49  424 
49  462 
49  5oo 

9.51  648 
9.5i  691 
9.5i  734 

0. 48  352 
0.48  309 
0.48266 

9 
9 
9 

97775 
97771 
97767 

49 

48 

47 

i4 
i5 
i6 

9 
9 
9 

49  539 
49577 
49615 

38 
38 

9.5i  776 
9.5i  819 
9.5i  861 

43 
42 

0.48  224 
0.48  181 
0.48  139 

9 
9 
9 

97763 
97759 
97754 

46 
45 
44 

'7 
i8 

'9 

9 
9 
9 

49654 
49  692 
49730 

39 
38 
38 
38 

38 
38 
38 
38 
38 
38 
38 
38 
38 
38 

9. 5 1  903 
9.5i  946 
9.51  988 

43 
42 
43 
42 
42 
42 

0.48  097 
0.48  o54 
0.48  012 

9 
9 
9 

97750 
97  746 
97  742 

43 
42 
4i 

40 

39 
38 
37 

20 

9 

49768 

9.52  o3i 

0.47  969 

9 

97  738 

21 
22 
23 

9 
9 
9 

49806 
49844 
49882 

9.52  073 
9.52  1x5 
9.62  i57 

0.47  927 
0.47885 
0.47843 

9 
9 
9 

97734 
97729 
97725 

24 
25 
26 

27 
28 
29 

9 
9 
9 

9 
9 
9 

49  920 
49958 
49996 

5oo34 

50  072 
5o  1 10 

9.52  200 
9.52  242 
9.52  284 

9.52  326 
9.52  368 
9.52  4io 

42 
42 
42 
42 
42 
42 

0.47  800 
0.47758 
0.47  716 

0.47  674 
0.47632 
0.47  590 

9 
9 
9 

9 
9 
9 

97721 
97717 
97713 

97708 
97  704 
97  700 

36 
35 
34 

33 

32 

3i 
30 

30 

9 

5o  i48 

9.52  452 

0.47548 

9 

97696 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

71 

°30 

1 

PP 

.1 

.2 

43 

4a 

39 

38 

.X 
.2 

5 

4 

4-3 
8.6 

4-2 

8.4 

.1 

.2 

11 

7.6 

OS 

I.O 

0.4 
0.6 

•3 

12.9 

12.6 

•3 

I 

1-7 

11.4 

•3 

i-S 

1.2 

•4 

17.2 
a5-8 

16.8 

21.0 

25.2 

•4 
•5 

.6 

I 
I 

3 

5.6 
9-5 
3-4 

IS-2 
19.0 
22.8 

■4 
•5 
.6 

2.0 

2-5 

30 

1.6 
2.0 

2-4 

:i 

30- 1 
34-4 

2^2— 

29.4 

33-6 

-.1 

2 
■ 

7-3 
1.2 

26.6 
30-4 

■7 
.8 

35 
4.0 

2.8 

66 


18° 

30. 

L.Sin.      d. 

L.  Tang,  j  d. 

L.  Cotg. 

L.  Cos.     d. 

30 

9.50  i48 

9 .52  452 

0.47548 

9.97696 

30 

3i 

9.60  i85 

38 

9.52  494 

42 

0.47  5o6 

9 .  07  69 1 

29 

32 

9.60  223 

38 

9.52  536 

0.47  464 

9.97687 

28 

33 

9.50  261 

9.52  578 

0.47  422 

9.97683 

4 

27 

37 

42 

4 

34 

9,60  298 

38 

9.62  620 

41 

0.47  38o 

9.97679 

5 

26 

3i) 

9.50  336 

38 

9.52  661 

0.47  339 

9.97674 

25 

36 

9.50  374 

9.52  703 

0.47  297 

9.97670 

24 

37 

42 

4 

■^7 

g.So  4i  I 

38 

9.52  745 

42 

0.47  255 

9.97  666 

23 

38 

9.50449 

9.52  787 

0.47  2l3 

9.97  662 

22 

39 

9.50  486 

37 

9.52  829 

42 

0.47  171 

9.97657 

^ 

21 

40 

9.5o  523 

38 
37 

9.52  870 

0.47 i3o 

9.97  653 

20 

4i 

9.5o  56i 

9.52  912 

0.47  088 

9.97649 

4 
4 

'9 

42 

g.So  598 

9.52  953 

0.47  047 

9.97645 

18 

43 

9.5o635 

37 
38 
37 

9.52  995 

42 

0.47  005 

9.97  640 

b 

17 

44 

9.50  678 

9.53  087 

0.46  963 

9.97686 

16 

At) 

9.5o  710 

9.53078 

0  46  922 

9.97  682 

i5 

46 

9.50  747 

37 
37 

9.53  120 

42 
41 

0.46  880 

9.97  628 

4 

S 

i4 

47 

9.50  784 

9.53  161 

0.46889 

9.97  628 

i3 

48 

9.50  821 

9.53  202 

0.46  798 

9.97619 

12 

49 

9.5o858 

37 
38 

37 

9.53  244 

42 
41 

42 

0.46  756 

9.97615 

4 
5 

4 

4 

II 
10 

50 

9.50  896 

9.53285 

0.46  715 

9.97  610 

5i 

9.50  933 

9.53  327 

0.46  673 

9.97  606 

9 

52 

9.5o  970 

9.53  368 

0.46  632 

9.97  602 

8 

53 

9.5i  007 

37 

9.53  409 

41 

0.46  591 

9.97597 

i 

7 

54 

9.5i  o43 

36 

9.53450 

41 

0.46  550 

9.97593 

4 

6 

55 

9.5i  080 

9.53  492 

o.46  5o8 

9.97589 

5 

56 

9.51  117 

37 

9.53533 

41 

0.46  467 

9.97  584 

b 

4 

57 

9. 5 1  1 54 

37 

9.53574 

0.46  426 

9.97  58o 

3 

58 

9.5i  191 

37 

9.53  6i5 

0. 46  385 

9.97576 

2 

59 

9.5i  227 

30 
37 

9-53  656 

41 
41 

0.46  344 

9.97571 

b 
4 

I 

60 

9.  5 1  264 

9.53  697 

0.46  3o3 

9.97567 

0 

L.  Cos. 

d. 

L.  Cotg.      d.  i  L.  Tang. 

L.Sin.    ;d.| 

' 

71°.                                                      1 

PR 

.1 

4* 

41 

38 

•  I 

37 

36 

.1 

5 

4 

4.2 

4.1 

3-8 

3-7 

3.6 

o-S 

°i 

.2 

8.4 

8.3 

7.6 

.2 

7-4 

7-2 

.2 

1.0 

0.8 

•3 

12.6 

12.3 

II. 4 

•3 

II. I 

10.8 

•3 

1-5 

1.2 

•4 

16.8 

16.4 

15.2 

•4 

14.8 

»4-4 

•4 

2.0 

X.6 

•5 

21.0 

20.5 

19.0 

•5 

18.5 

18.0 

•5 

2-5 

2.0 

.6 

252 

24.6 

22.8 

.6 

22.2 

21.6 

.6 

30 

2.4 

■7 

29.4 

28.7 

26.6 

•7 

25-9 

25.2 

•7 

3-5 

2.8 

.8 

336 

32.8 

30-4 

.8 

29.0 

28.8 

.8 

4.0 

3-2 

■?   '     r-' 

3.6 

67 


19°. 


0 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.     d.  1       1 

9. 5 1  264 

37 
37 
36 

9.53  697 

0.46  3o3 

9.97  567 

60 

I 

2 

3 

9.51  3oi 
9.5i  338 
9.51  374 

9.53  738 
9.53779 
9.53820 

0.46  262 
0.46  221 
0.46  180 

9.97  563 
9.97  558 
9.97  554 

5        59 

58 

*      57 

4 
5 
6 

9.51  4ii 
9.5i  447 
9.61  484 

36 

37 
16 

9.53  86i 
9.53  902 
9.53  943 

0.46  139 
0.46  098 
0.46  067 

9.97550 
9.97  545 
9.97  541 

,      56 

55 

*      54 

7 
8 

9 

9.61  520 
9.5i  557 
9.61  593 

37 

36 
36 

37 
36 
36 
36 
37 
36 
36 
36 
35 
36 

36 
36 
36 
36 
36 
36 

9.53984 
9.54  025 
9.54065 

0.46  016 
0.45  975 
0.45  935 

9.97  536 
9.97  532 
9.97  528 

I      53 

52 

'      5i 

10 

9.61  629 

9.54  106 

0.45  894 

9.97  523 

50 

1 1 

12 

i3 

9.61  666 
9.51  702 
9.51  738 

9.54  i47 
9.54  187 
9.54  228 

0.45  853 
0.45  8i3 
0.45  772 

9-97519      ^ 
9.97515 
9.97  5io 

,      49 
48 

'      47 

i4 
i5 
i6 

9-5i  774 
9.61  811 
9.51  847 

9.54  269 
9.54  309 
9.54  350 

0.45  73i 
0.45  691 
0.45  65o 

9.97  5o6 
9.97  5oi 
9.97497      ^ 

'      46 
45 
44 

I? 
i8 

'9 

9.5i  883 
9.61  919 
9.61  955 

9.54  390 
9.54  43 1 
9.54  471 

0.45  610 
0.45  569 
0.45  529 

9.97492      " 
9-97  488 
9-97  484      ^ 

43 
42 

4i 

20 

9.51  991 

9.54  5i 2 

0.45488 

9-97479 

40 

21 
22 
23 

9.52  027 
g.52  o63 
9.62  099 

9.54552 
9.54593 
9.54  633 

0.45448 
0.45  407 
0.45  367 

9-97475      = 
9-97470      " 
9.97466 

39 

38 
37 

24 
25 
26 

9.52  135 
9.52  171 
9.52  207 

9.54673 
9.54  714 
9.54754 

0.45  327 
0.45  286 
0.45  246 

9.97461       ■ 
9.97457 
9.97453      ^ 

36 
35 
34 

27 
28 
29 

9.52  242 
9.52  278 
9.52  3i4 

35 
36 

36 
36 

9.54  794 
9.54  835 
9.54875 

0.45  206 
0.45  i65 

0.45   125 

9.97448      ^ 
9.97  444 
9.97439     - 

33 

32 

3i 

30 

9. 62  350 

9.54915 

0.45  o85 

.9.97435 

30 

L.  Cos. 

d. 

L.  Cotg.  1   d.  1 

L.  Tang. 

L.  Sin.     d 

1~ 

70°  30 .                                                  1 

PF 

■  I 
.3 

•  3 

»         41 

40 

37 

.1 

.2 

•3 

36 

3-6 
7.2 
10.8 

35 

3-5 
7.0 
10.S 

.1 

.2 

•3 

5 

4 

4-1 
8.2 
12.3 

4.0 
8.0 
12.0 

3-7 
7-4 
II. I 

0.5 

I.O 

1-5 

0.4 
0.8 
1.2 

•  4 

•  5 

.6 

16.4 
20.5 
24.6 

16.0 
20.0 
24.0 

148 
18.S 
22.2 

•4 
•5 
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14-4 
18.0 

21.6 

14.0 
17-5 
21.0 

•4 
•5 
.6 

2.0 

2-5 

30 

1.6 
2.0 

2-4 

■7 

.8 

28.7 
32.8 

28.0 
32.0 
36.0 

25-9 
29.0 

33,3 

•7 
.8 

25.2 
28.8 

245 
28.0 

•7 
.8 

3-5 
4.0 

2.8 

3-2 

3.6 

68 


19° 

30. 

/ 

L.  Sin. 

d. 

L.  Tang. 

d.   1   L.  Cotg. 

L.  Cos.    :  d. 

30 

9.52  350 

35 

^6 

9.54  915 

40 
40 

0.45  oa5 

9.97435 

30 

3i 

9.52  385 

9.54  955 

0.45  045 

9 

97  43o 

4 

29 

32 

9.52  421 

3S 
36 

9.54  995 

40 
40 

0.45  005 

9 

97  426 

28 

33 

9.52  456 

9.55  o35 

0.44  965 

9 

97  421 

4 

27 

34 

9.52  492 

35 

9.55  075 

40 

0.44  925 

9 

97417 

5 

26 

35 

9.52  527 

36 

9.55  1 15 

0.44  885 

9 

97  4i2 

2b 

36 

9.52  563 

9.55  i55 

0.44845 

9 

97  4o8 

24 

35 

40 

.S 

3? 

9.52  598 

36 

9.55  195 

0.44  805 

9 

97  4o3 

4 

23 

38 

9.52  634 

9.55235 

0.44  765 

9 

97  399 

22 

39 

9.52  669 

36 

9.55  275 

0.44  725 

9- 

97  394 

4 

5 

2  1 

40 

9.5«  705 

9.55  315 

0.44  685 

9- 

97  390 

20 

40 

4i 

9.52  740 

9.55  355 

0.44  645 

9- 

97  385 

4 

'9 

42 

9.52  775 

36 

9.55395 

0.44  6o5 

9- 

97  38i 

18 

43 

9.52  811 

9.55434 

39 

0.44  566 

9- 

97  376 

17 

35 

40 

4 

44 

9.52  846 

9.55  474 

0.44  526 

9- 

97372 

5 

16 

45 

9.52  88i 

9.55514 

0.44  486 

9- 

97  367 

i5 

46 

9.52  916 

35 
35 

9.55554 

40 
39 

0.44  446 

9- 

97  363 

5 

i4 

47 

9.52  95r 

9.55  593 

40 

0.44  407 

9- 

97  358 

■; 

i3 

4>^ 

9.52  986 

9.55  633 

0.44  367 

9- 

97  353 

12 

49 

9.53  021 

35 

9.55673 

40 

0.44  327 

9- 

97  349 

1 1 

50 

9.53  o56 

36 

9.55  712 

0.44288 

9- 

97  344 

10 

5i 

9.53  092 

9.55  752 

39 

0.44248 

9- 

97  34o 

s 

9 

02 

9.53  126 

9.55  791 

0.44  209 

9- 

97  335 

8 

53 

9.53  i6i 

35 

35 

9.55  83i 

39 

0.44  169 

9- 

97  33i 

5 

7 

54 

9.53  196 

35 

9.55870 

0.44  i3o 

9- 

97  326 

4 

6 

55 

9.53  2  3 1 

9.55  910 

0.44  090 

9- 

97   322 

5 

5 

56 

9.53  266 

35 

9.55  949 

39 

0.44  o5 1 

9- 

97  317 

4 

35 

40 

5 

^7 

9.53  3oi 

35 

9.55  989 

39 

0.44  on 

9- 

97  3l2 

4 

3 

58 

9.53  336 

9.56  028 

0.43  972 

9- 

97  3o8 

5 

2 

59 

9.  53  370 

34 

9.56  067 

39 

0.43  933 

9- 

97  3o3 

I 

35 

40 

60 

9.53405 

9.56  107 

0.43893 

9- 

97  299 

0 

L.  Cos. 

d. 

L.  Cotgr.  1   d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

70°. 

1 

PP 

.1 

40 

4.0 

— 

39 

3-9 

36 

3-6 

.1 

35 

34 

.1 

5 

4 

35 

3-4 

0.5 

°i 

.2 

8.0 

7.8 

7-^ 

.2 

7.0 

6.8 

.3 

I.O 

0.8 

•3 

12.0 

>-7 

10.8 

•3 

10.5 

10.2 

•3 

1-5 

1.8 

•4 

16.0 

S.6 

14.4 

■4 

140 

13.6 

•4 

a.o 

1.6 

•s 

20.0 

9-5 

18.0 

•5 

17- S 

17.0 

•s 

as 

2.0 

.6 

24.0 

!3-4 

21.6 

.6 

21.0 

20.4 

.6 

3° 

2.4 

•7 

28.0 

87-3 

25.2 

•7 

24  5 

23.8 

•7 

3-5 

2.8 

.8 

32.0 

51.2 

28.8 

.8 

28.0 

27.2 

.8 

4.0 

3-2 

'            '^ 

36.0 

_^ 

^5J_ 

30.. 

^^^ 

^ 

45 

«il- 

69 


20°. 


L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos.     d. 

0 

9.53405 

g.5b  107 

0.43  893 

9.97299 

60 

I 

9.53440 

35 

9.56  i46 

39 

0.43  854 

9- 

97294 

5 

59 

2 

9.53475 

9.56  i85 

0.43815 

9- 

97  289 

58 

3 

9.53  509 

35 

9.56  224 

39 
40 
39 

0.43776 

9- 

97285 

4 

57 

4 

9.53544 

9.56  264 

0.43736 

9- 

97  280 

4 

56 

5 

9.53578 

9.56  3o3 

0.43697 

9- 

97276 

55 

6 

9.53  6i3 

35 
34 

9.56  342 

39 

0.43  658 

9- 

97271 

S 

54 

7 

9.53  647 

35 

9.56  381 

39 

0.43  619 

9- 

97  266 

4 

53 

8 

9.53682 

9.56  420 

0.43  58o 

9- 

97  262 

52 

9 

9.53716 

34 

9.56  459 

39 

0.43541 

9- 

97257 

5 

5i 

10 

9.53751 

9.56498 

39 
39 

0.43  5o2 

9- 

97  252 

50 

II 

9.53785 

34 

9.56  537 

0.43  463 

9- 

97  248 

4 
5 

49 

12 

9.53819 

9.56576 

39 

0.43  424 

9- 

97243 

48 

i3 

9.53  854 

9.566i5 

0.43385 

9- 

97  238 

5 

47 

34 

39 

4 

i4 

9.53  888 

9.56  654 

39 

0.43346 

9- 

97234 

46 

i5 

9.53  922 

9.56693 

0.43  307 

9- 

97229 

45 

i6 

9.53957 

35 
34 

9.56782 

39 

0.43268 

9- 

97  224 

b 

4 

44 

17 

9.53991 

9.56771 

39 

0.43  229 

9- 

97  220 

5 

43 

i8 

9.54  025 

9.56  810 

0.43  190 

9- 

97  2l5 

42 

'9 

9.54  059 

34 

9.56  849 

38 

0.43  i5i 

9- 

97  210 

5 

4i 

20 

9.54  093 

34 

9.56887 

0.43  1 13 

9- 

97  206 

4 

40 

21 

9.54  127 

34 
34 

9.56  926 

39 

0.43  074 

9- 

97  201 

5 

5 

39 

22 

9.54  161 

9.56965 

0.43  o35 

9- 

97  196 

38 

23 

9.54  195 

34 
34 

9.57  oo4 

38 

0.42  996 

9- 

97  192 

4 

37 

24 

9.54  229 

34 

9.57  o42 

39 

0.42  958 

9- 

97  187 

36 

25 

9.54263 

9.57  081 

0.42  919 

9- 

97  182 

35 

26 

9.54297 

34 
34 

9.57  120 

38 

0.42  880 

9- 

97178 

4 
5 

34 

27 

9.54  33i 

9.57  i58 

39 

0.42  842 

9. 

97173 

5 

33 

28 

9.54  365 

9.57  197 

38 

0.42803 

9- 

97  168 

32 

29 

9.54  399 

34 

9.57  235 

0.42  765 

9- 

97  1 63 

5 

3i 

34 

4 

30 

9.54433 

9.57  274 

0.42  726 

9- 

97  159 

30 

L.  Cos. 

d. 

L.Cotgr.  1  d. 

L.  Tang. 

L.  Sin. 

d. 

' 

69°  30 

', 

1 

PP 

.1 

40 

39 

38 

.1 

35 

34 

.1 

5 

4 

4.0 

It 

3-8 

V5 

3-4 

0-5 

0.4 

.2 

8.0 

7.6 

.2 

J.O 

6.8 

.2 

1.0 

0.8 

•3 

12. 0 

'•7 

11.4 

•3 

i( 

5-5 

ia2 

•3 

1-5 

1.2 

•4 

16.0 

5-6 

»5-2 

•4 

li 

i-o 

t3-6 

•4 

2.0 

1.6 

•5 

20.0 

95 

19.0 

•5 

I 

r-5 

17.0 

5 

2-5 

2.0 

.6 

24.0 

'3- 4 

22.8 

.6 

2 

.0 

20.4 

.6 

30 

2-4 

•7 

28.0 

»7-3 

26.6 

•7 

2i 

(-5 

238 

7 

3-5 

2.8 

.8 

32.0 

)1.2 

30.4 

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2 

i.o 

27.2 

8 

4.0 

3-2 

36.0 

^5J_ 

■  5 

30.6 

^ 

'■'        ,T^    1 

70 


20° 

30 

'. 

L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos.     d. 

30 

9.54433 

9.57  274 

38 

0.42  726 

9.97  159 

30 

3i 

9.54466 

9.57  3l2 

0.42  688 

9- 

97  1 54 

29 

32 

9.54  5oo 

9.57  35i 

38 

0.42  649 

9- 

97  149 

28 

33 

9.54534 

9.57  389 

0.42  611 

8. 

97  145 

27 

33 

39 

S 

34 

9.54567 

9.57428 

38 

0.42  572 

9- 

97  i4o 

5 

26 

35 

9.54  601 

9.57  466 

38 
39 
38 

0.42  534 

9- 

97135 

25 

36 

9.54635 

34 

9.57  5o4 

0.42  496 

9- 

97  i3o 

5 

24 

37 

9.54668 

9.57  543 

0.42  457 

9- 

97  126 

s 

23 

38 

9.54  702 

9.57  58i 

38 

0.42  419 

9- 

97  121 

22 

39 

9.54735 

33 

9.57  619 

0.42  38i 

9- 

97  116 

5 

21 

34 

39 

40 

9.54  769 

33 

9.57658 

38 
38 

0.42  342 

9- 

97  III 

4 

20 

•9 

4i 

9.54  802 

9.57  696 

0.42  3o4 

9- 

97  107 

42 

9.54  836 

9.57734 

38 

38 
39 
38 
38 
38 
38 
38 

0.42  266 

9- 

97  102 

18 

43 

9.54869 

33 

9.57772 

0.42  228 

9- 

97097 

b 

17 

44 

9.54903 

34 
33 

9.57  810 

0.42  190 

9- 

97092 

16 

45 

9.54936 

9.57849 

0.42  i5i 

9- 

97087 

i5 

46 

9.54969 

33 

9.57887 

0.42  ii3 

9- 

97083 

i4 

47 

9.55  oo3 

34 
33 

9.57925 

0.42  075 

9 

97078 

i3 

48 

9.55  o36 

9.57963 

0.42  037 

9- 

97073 

12 

49 

9.55  069 

33 
33 

9.58  001 

o.4i  999 

9 

97  068 

5 

S 

1 1 

50 

9.55  102 

9.58  039 

38 
38 

0.41  961 

9 

97  o63 

10 

5i 

9.55  i36 

34 
33 

9.58  077 

o.4i  923 

9 

97059 

5 

9 

52 

9.55  169 

9.58115 

o.4i  885 

9 

97054 

8 

53 

9.55  202 

33 

9.58  i53 

38 
38 
38 

o.4i  847 

9 

97049 

b 

7 

54 

9.55  235 

9.58  191 

o.4i  809 

9 

97  o44 

5 

6 

55 

9.55268 

9.58  229 

o.4i  771 

9 

97039 

5 

56 

9.55  3oi 

33 

9.58  267 

38 

o.4i  733 

9 

97035 

4 

4 

57 

9.55334 

33 

9.58  3o4 

37 
38 

0.4 1  696 

9 

97  o3o 

3 

58 

9.55  367 

9.58  342 

o.4i  658 

9 

97025 

2 

59 

9.55  4oo 

33 
33 

9.58  38o 

38 
38 

o.4i  620 

9 

97  020 

b 
5 

I 
0 

60 

9.55433 

9.584i8 

0.41  582 

9 

97  01 5 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin.    1  d. 

' 

69°.                                                     1 

PP 

.1 

39 

38 

37 

.1 

34 

33 

.1 

5 

4 

3-9 

3.8 

3-7 

V4 

3-3 

0.5 

0.4 

.2 

7.8 

7.6 

7-4 

.2 

< 

i.8 

6.6 

.2 

I.O 

0.8 

•3 

11.7 

11.4 

II. I 

•3 

K 

}.3 

9.9 

•3 

I- 5 

1.2 

•4 

15.6 

IS-2 

14.8 

•4 

I 

,.6 

13.2 

•4 

3.0 

1.6 

•S 

19-5 

19.0 

18.5 

•5 

I 

7.0 

16.5 

•5 

2-5 

2.0 

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234 

22.8 

22.2 

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2 

'•4 

19.8 

,6 

30 

2.4 

•7 

27.3 

26.6 

25  9 

•7 

3 

5.8 

23- 1 

7 

35 

2.8 

.8 

31.2 

30-4 

29.6 

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2 

7-2 

26.4 

.8 

4.0 

32 

_J5-     1 

312. 

X-^ 

3.6     1      29.7     1 

-^^ 

71 


2r 


' 

L.Sin.    1   d.J 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.55433 

9.584i8 

0.41  582 

9.97  oi5 

60 

I 

9.55466 

33 

9.58  455 

37 
38 

o.4i  545 

9 

97  010 

5 

59 

2 

9.55  499 

9.58  493 

o.4i  507 

9 

97  oo5 

58 

3 

g.55  532 

32 

9.58  53i 

38 
38 

0.4 1  469 

9 

97  001 

5 

57 

4 

9.55  564 

9.58  569 

37 

o.4i  43i 

9 

96  996 

5 

5b 

5 

9.55  597 

9.58  606 

0.4 1  394 

9 

96991 

55 

6 

9.55  63o 

33 
33 

9.58  644 

38 
37 

0.41  356 

9 

96986 

5 

5 

54 

7 

9.55  663 

9.58681 

38 

o.4i  319 

9 

96  981 

s 

53 

8 

9.55  695 

9-58  719 

o.4i  281 

9 

96976 

52 

9 

9.55  728 

33 

9.58757 

38 

o.4i  243 

9 

96971 

5i 

10 

9.55  761 

9.58794 

37 

0.41  206 

9 

96  966 

50 

1 1 

9. 55  793 

32 

9.58  832 

38 

o.4i  168 

9 

96  962 

49 

12 

9.55826 

9.58869 

37 

0.41  i3i 

9 

96957 

48 

i3 

9.55  858 

32 

9.58  907 

38 

o.4i  093 

9 

96  952 

47 

i4 

9.55  891 

33 

9.58944 

37 

o.4i  o56 

9 

96947 

5 

46 

i5 

9.55  923 

9-58  981 

37 

0.41  019 

9 

96  942 

45 

i6 

9.55  956 

33 

9.59  019 

38 

o.4o  981 

9 

96937 

44 

I? 

9.55988 

32 

9.59  o56 

37 
38 

o.4o  944 

9 

96  932 

b 

43 

i8 

9.56  021 

9.59  094 

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9 

96  927 

42 

'9 

9-56  o53 

32 
32 

33 

9.59  i3i 

37 
37 

37 
38 

o.4o  869 

9 

96  922 

5 

5 
5 

4i 
40 
39 

20 

9.56  o85 

9.59  168 

o.4o  832 

9 

96917 

21 

9.56  118 

9.59  2o5 

0.40  795 

9 

96  912 

2  2 

9.56  i5o 

9.59  243 

0.40.  757 

'9 

96907 

38 

23 

9.56  182 

32 

9.59  280 

37 

0.40  720 

9 

96  903 

37 

24 

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33 

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37 

0.40  683 

9 

96898 

5 

36 

25 

9.56  247 

9.59354 

37 

o.4o  646 

9 

96893 

35 

26 

9-56  279 

32 

9.59  391 

37 

0.40  609 

9 

96888 

34 

27 

9.56  3i I 

32 

9.59429 

38 

o.4o  571 

9 

96883 

b 
5 

33 

28 

9.56  343 

9.59  466 

37 

0.40  534 

9 

96  878 

32 

29 

9.56  375 

32 

9.59  5o3 

37 
37 

o.4o  497 

9 

96873 

5 

3i 
30 

30 

9.56  4o8 

33 

9.59  540 

o.4o  460 

9 

96868 

L.  Cos.    1  d.  1 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

6J 

i°3( 

1 

PP 

.1 

38 

37 

.1 

33 

3a 

.1 

5 

4 

3.8 

3-7 

3-3 

3-2 

0.5 

0-4 

•  2 

7.6 

7-4 

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6.6 

6.4 

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I.O 

0.8 

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II. 4 

II. I 

•3 

9-9 

9.6 

•3 

IS 

I. a 

•4 

15.2 

14.8 

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I 

3-2 

12.8 

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a.o 

1.6 

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19  0 

18.5 

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I 

5.5 

16.0 

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a.o 

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22.8 

22.2 

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I 

J.  8 

19.3 

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3.0 

a- 4 

•7 

26.6 

25.9 

■7 

2 

31 

22.4 

•7 

3-5 

28 

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30-4 

29.6 

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2 

t..4 

25.6 

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4.0 

3-2 

•9  '       34-' 

U— 

28.8 

^^ 

^ il— 

I' 

72 


21°  30. 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

^^^^ 

30 

9.56  4o8 

32 
32 
32 

9. 59  540 

37 
37 
37 

0.40  460 

9.96  868 

5 
5 

5 

30 

29 
28 
27 

3i 

32 

33 

9.56  44o 
9-56  472 
9.56  5o4 

9.59577 
9.59  6i4 
g.59  65i 

0.40  423 
o.4o  386 
o.4o  349 

9 
9 
9 

96  863 
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34 
35 
36 

9.56  536 
9.56  568 
9.56  599 

32 
32 

3' 

9.59688 
9.59725 
9.59  762 

37 
37 
37 

o.4o  3l2 
0.40  275 
0.40  238 

9 
9 
9 

96848 
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96  838 

b 
5 
5 

26 

25 

24 

3? 
38 
39 

9.56  63i 
9.56  663 
9-56  695 

32 
32 
32 

32 
31 
32 
32 
32 
31 
32 
31 
32 
32 
31 
32 
31 

9.59799 
9.59  835 
9.59  872 

37 
36 
37 
37 

37 
37 
36 

37 
37 
37 
36 
37 
37 
36 

37 
36 

37 
36 
37 
36 

0.40  201 
0.40  165 
o.4o  128 

9 
9 
9 

96833 
96828 
96  823 

b 
5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 
6 

5 

S 

5 
5 

23 
22 
21 

20 

'9 

18 

17 

16 
i5 
i4 

i3 
12 
I  I 

10 

9 

8 

7 

40 

9.56  727 

9.59909 

o.4o  091 

9 

96818 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

9.56  759 
g.56  790 
9-56  822 

9.56  854 
9.56  886 
9.56  917 

9-56  949 

9.56  980 

9.57  012 

9.59  946 

9.59  983 

9.60  019 

9.60  o56 
9.60  093 
9.60  i3o 

9.60  166 
9.60  2o3 
9.60  24o 

o.4o  o54 
o.4o  017 
0.39  981 

0.39  944 
0.39  907 
0.39  870 

0.39  834 
0.39  797 
0.39  760 

9 
9 
9 

9 
9 
9 

9 
9 
9 

96813 
96  808 
96  8o3 

96798 
96793 
96  788 

96  783 
96  778 
96  772 

50 

9.57  o44 

9.60  276 

0.39  724 

9 

96  767 

5i 

52 

53 

9.57  075 
9.57  107 
9.57  i38 

9.60  3i3 
9.60  349 
9.60  386 

0.39  687 
0.39651 
0.39  6i4 

9 
9 
9 

96  762 
96757 
96  752 

54 
55 
56 

9.57  169 
9.57  201 

9.57  232 

32 
31 

9.60  422 
9.60  459 
9.60  495 

0.39  578 
0.39  54 1 
0.39  505 

9 
9 
9 

96  747 
96  742 
96  737 

5 

5 

6 
5 
4 

5? 
58 
59 

9.57  264 
9.57295 
9.57  326 

31 
3' 
32 

9.60  532 
9.60  568 
9.60  605 

36 
37 
36 

0.39468 

0.39  432 
0.39  395 

9 
9 
9 

96  732 
96  727 
96  722 

5 

5 

5 

3 
2 
I 

0 

60 

9.57  358 

9.60  64 1 

0. 39  359 

9 

96  717 

L.  Cos. 

d. 

L.  Cotg-. 

d. 

L.  Tang. 

L.  Sin. 

d. 

68°.                                                         1 

PP 

.1 

■  2 

•3 

37 

36 

.1 

.2 
•3 

32 

31 

.1 
.2 

•3 

6 

5 

3-7 
7-4 
11. 1 

3-6 

7-2 

10.8 

3-2 

6.4 
9.6 

3-1 
6.2 

9-3 

0.6 
1.2 
1.8 

0-5 
1.0 

•4 
•5 
.6 

.4.8 
18.S 
22.2 

14.4 
18.0 
21.6 

•4 
•5 

.6 

12.8 
16.0 
19.2 

12.4 
15-5 
18.6 

•4 
■5 
.6 

2-4 

30 

3.6 

2.0 

2.5 
3.0 

-.1 

25.9 
29.6 

25.2 
28.8 

•7 
.8 

22.4 
25.6 

28.8 

21.7 
24.8 

27..2_ 

•7 
.8 

4-2 

4.8 

3-5 
4.0 

73 


22°, 


L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

9.57  358 

9.60  64i 

36 
37 

0.39  359 

9.96  717 

60 

1 

9.57  38^ 

31 

9.60  677 

0.39  323 

9.96  71 1 

59 

2 

g.57  420 

9.60  714 

36 
36 

0.39  286 

9.96  706 

58 

■i 

9.57  45i 

3> 

9.60  750 

0.39  25o 

9.96  701 

,5 

57 

4 

9.57482 

32 

9.60  786 

37 

0.89  2l4 

9.96  696 

56 

b 

9.57  5i4 

9.60823 

36 

0.39   177 

9.96  691 

55 

6 

9.57545 

9.60  859 

0.39  i4i 

9.96686 

54 

3' 

36 

s 

7 

9.57  576 

9.60  895 

36 

0.39  105 

9.96  681 

53 

8 

9.57  607 

9.60  931 

36 
37 
36 
36 

0.39  069 

9.96  676 

52 

9 

9.57  638 

31 
31 
3' 

9.60  967 

0.39  o33 

9.96  670 

5 
5 

50 

49 

10 

1 1 

9.57  669 

9.6t  oo4 

0.38  996 

9.96  665 

9.57  700 

9.61  o4o 

0.  38  960 

9.96  660 

12 

9.57  731 

9.61  076 

%6 

0.38  924 

9.96655 

48 

i3 

9.57  762 

9.61  1 12 

36 

0. 38  888 

9.96  650 

5 

47 

i4 

9.57793 

3' 

9.61  i48 

36 

0. 38  852 

9.96  645 

46 

lb 

9   57  824 

9.61  i84 

36 
36 

0. 38  816 

9.96  64o 

45 

i6 

9.57  855 

30 

9.61  220 

0.38  780 

9.96  634 

s 

44 

17 

9.57885 

3' 

9.61  256 

36 

0.38  744 

9.96  629 

43 

i8 

9.57  916 

9.61  292 

36 
36 

36 
36 
36 
36 
36 

0.38  708 

9.96  624 

42 

'9 

9.57  947 

31 

9.61  328 

0.38  672 

9.96  619 

5 

5 
6 

4i 
40 

20 

9.57  978 

9.61  364 

0. 38  636 

9.96  6i4 

21 

9.58008 

9.61  4oo 

0.38  600 

9.96  608 

39 

22 

9.58  039 

9.61  436 

0. 38  564 

9.96  6o3 

38 

23 

9.58  070 

3' 

9.61  472 

0.38  528 

9.96  598 

b 

37 

24 

9-58  loi 

9.61  5o8 

0.38492 

9.96  593 

5 

36 

2b 

9-58  i3i 

9.61  544 

0. 38  456 

9.96  588 

35 

26 

9-58  162 

31 

9.61  579 

35 
36 
36 

0.38  42I 

9.96  582 

34 

27 

9-58  192 

30 
3' 

9.61  6i5 

o,38  385 

9.96577 

5 

5 

33 

28 

9.58  223 

9.61  65i 

36 

0.38  349 

9.96  572 

32 

29 

9.58253 

30 

9.61  687 

0.38  3i3 

9  .96  567 

i 

3i 

30 

9-58  284 

3' 

9.61  722 

0.38  278 

9.96  562 

b 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin.     d.| 

' 

67°  30 .                                                    1 

PP 

I 

37 

36 

35 

I 

3a 

31 

30 

.1 

6 

a6 

5 

05 

3-7 

3.6 

35 

1' 

3« 

30 

.2 

74 

7-2 

7.0 

.3 

64 

6.3 

6.0 

3 

I  2 

I.O 

3 

II. I 

108 

I 

3.5 

•3 

9.6 

9-3 

9.0 

•3 

1.8 

1-5 

•4 

14.8 

14-4 

I 

♦  0 

-4 

12.8 

12.4 

12.0 

•4 

24 

2  0 

•  5 

18.S 

18.0 

I 

7-S 

•5 

16.0 

'55 

150 

•5 

30 

2-5 

.6 

22.2 

21.6 

3 

1.0 

6 

19.2 

18.6 

18.0 

.6 

36 

30 

•7 

25  9 

25-a 

a 

♦  ■) 

•7 

33  4 

21  7 

21.0 

.7 

42 

3-5 

.8 

29.6 

28.8 

3 

>io 

.8 

25.6 

248 

24.0 

.8 

48 

40 

-^ 

.9              28.8        1         27   Q        1         27.0 

— SL 

74 


22 

°30 

'• 

/ 

L.  Sin.       d. 

L.  Tang.  |  d.  |  L.  Cotg. 

L.  Cos. 

d. 

30 

9.58  284 

9.61  722 

36 
36 

0.38  27a 

9.96  562 

(, 

30 

3i 

9-58  3i4 

30 

9.61  758 

0.38  242 

9- 

96  556 

s 

29 

32 

9.58  345 

9.61  794 

0.38  206 

9 

96551 

28 

33 

9-58  375 

30 

9.61  83o 

30 

0.38  170 

9 

96546 

27 

34 

9.58  4o6 

31 

9.61  865 

3S 

36 

0.38  135 

9 

96541 

6 

26 

35 

9.58  436 

9.61  901 

0.38  099 

9 

96535 

25 

36 

9.58467 

3' 
30 

9.61  936 

35 
36 

0.38  064 

9 

96  53o 

5 

24 

;^7 

9-58  497 

9,61  972 

■?6 

0.38  028 

9 

96525 

S 

23 

38 

9-58  527 

9.62  008 

0.37  992 

9 

96  520 

6 

22 

39 

9-58  557 

9.62  o43 

35 
36 

0.37  957 

9 

96  5i4 

21 

40 

9.58  588 

31 

9.62  079 

0.37  921 

9 

96  509 

20 

4i 

9.58618 

30 

9.62  1 14 

35 
36 

0.37  886 

9 

96  5o4 

6 

'9 

42 

9.58  648 

9.62  150 

0.37  85o 

9 

96498 

18 

43 

9.58678 

30 
31 

9.62  i85 

35 
36 

0.37  815 

9 

96  493 

5 

17 

44 

9.58  709 

9.62  221 

0.37779 

9 

96488 

s 

16 

45 

9.58  739 

9.62  256 

36 

0.37  744 

9 

96  483 

6 

i5 

46 

9.58  769 

30 

9.62  292 

0.37  708 

9 

96477 

i4 

30 

35 

5 

47 

9.58799 

9.62  327 

35 

0.37  673 

9 

96  472 

s 

i3 

48 

9.58  829 

9.62  362 

0.37  638 

9 

96  467 

d 

12 

49 

9.58  859 

30 

9.62  398 

36 

0.37  602 

9 

96  46i 

1 1 

50 

9.58  889 

9.62  433 

0.37  567 

9 

96456 

10 

5i 

9.58  919 

9.62  468 

36 

0.37  532 

9 

96451 

6 

9 

52 

9.58  949 

9.62  5o4 

^ 

0.37  496 

9 

96445 

8 

53 

9.58979 

30 
30 

9.62  539 

35 
35 

0.37  461 

9 

96  44o 

5 

7 

54 

9.59  009 

9.62  574 

35 
36 

0.37  426 

9 

96  435 

6 

6 

55 

9.59  039 

9.62  609 

0.37  391 

9 

96  429 

5 

5 

56 

9.59  069 

3" 

9.62  645 

0.37  355 

9 

96  424 

4 

29 

35 

5 

5? 

9.59  098 

30 

9.62  680 

35 

0.37  320 

9 

.96  419 

6 

3 

58 

9.59  128 

9.62  715 

0.37285 

9 

96  4i3 

2 

59 

9.59  i58 

30 

9.62  750 

35 

0.37  250 

9 

.96408 

I 

60 

9.59  188 

3° 

9.62  785 

35 

0.37  215 

9 

.96  4o3 

0 

L.  Cos.       d. 

L.  Cotg.      d.     L.  Tang. 

L.  Sin. 

d. 

/ 

67°.                                                       1 

PP 

36 

35 

31 

30 

39 

6 

5 

.1 

3-6 

3-5 

3-' 

.1 

3-0 

2.9 

.1 

0.6 

0.5 

.2 

72 

7.0 

6.2 

2 

6.0 

5-8 

.2 

1.2 

1.0 

•3 

10.8 

JO.  5 

9-3 

■3 

9.0 

8.7 

-3 

1.8 

•  5 

•4 

M  4 

14.0 

12.4 

4 

12.0 

116 

■4 

2.4 

2.0 

•5 

18.0 

'7-5 

15-5 

•5 

J5.0 

14.5 

•5 

3-0 

25 

.6 

21.6 

21  0 

18.6 

6 

18.0 

I7-4 

.6 

3-6 

3-0 

■7 

25.2 

24.5 

21.7 

-7 

21.0 

20.3 

-7 

4.2 

3-5 

.8 

28.8 

28.0 

24-8 

.8 

24.0 

23.2 

.8 

4.8 

40 

Q    1        27.0       1      26.  I         1 

Q  1       ';.4      1      4-S      1 

75 


4 

2, 

3°. 

• 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.      d. 

0 

9.69  i»8 

9.62  7»5 

0.37215 

9.95  4o3 

6 

60 

I 

9.59  218 

9.62  820 

35 

0.37  180 

9.96397 

59 

2 

9.59  247 

9.62  855 

0.37  145 

9.96  392 

58 

3 

9.59277 

30 

9.62  890 

36 

0.37  no 

9.96  387 

6 

^7 

4 

9.59  307 

9.62  926 

3S 

0.37074 

9.96381 

56 

b 

9.59  336 

9.62  961 

0.37  039 

9.96  376 

6 

55 

6 

9.59  366 

30 

9.62  996 

0.37  oo4 

9.96  370 

54 

30 

35 

S 

7 

9.59  396 

29 

9.63  o3i 

0.36  969 

9.96365 

53 

8 

9.59  425 

g.63  066 

0.36  934 

9.96  36o 

6 

52 

9 

9.59455 

30 

9.63  101 

35 

0  36  899 

9.96354 

5i 

10 

9.59484 

29 

9.63  i35 

0.36  865 

9.96  349 

5 
6 

50 

1 1 

9.  59  5i4 

30 
29 

9.63  170 

0.36  83o 

9.96343 

49 

12 

9.59  543 

9.63  2o5 

0.36  795 

9.96338 

48 

i3 

9.59  573 

30 
29 

9.63  240 

35 
35 

0.36  760 

9.96333 

6 

47 

i4 

9.59  602 

9.63  275 

35 

0.36  725 

9.96  327 

5 

4b 

i5 

9.59  632 

9.63  3io 

0.36  690 

9.96  322 

45 

i6 

9.59  661 

29 

9.63345 

35 

0. 36  655 

9.96  3i6 

44 

I? 

9.59  690 

29 

9.63  379 

34 

0. 36  621 

9.96  3i  I 

5 
6 

43 

i8 

9.59  720 

9.63414 

0. 36  586 

9.96  3o5 

42 

'9 

9.59749 

29 

9.63  449 

35 

0.36  55I 

9.96  3oo 

5 
6 

4i 

29 

35 

20 

9.59778 

9.63  484 

0.36  5i6 

9.96  294 

40 

21 

9.59  808 

30 
29 

9.63  519 

0.36  48I 

9.96  289 

5 
5 
6 

39 

22 

9.59837 

9.63  553 

0.36  447 

g.96  284 

38 

23 

9.59866 

29 

9.63  588 

3.-> 

0.364I2 

9.96  278 

37 

29 

35 

S 

24 

9.59  895 

29 

9.63  623 

34 

0.36  377 

9.96  273 

6 

36 

2b 

9.59924 

9.63  657 

0. 36  343 

9.96  267 

35 

26 

9.59954 

30 
29 

9.63  692 

35 
34 

0.36  3o8 

9.96  262 

5 
6 

34 

27 

9.59  983 

29 

9.63  726 

35 

0.36  274 

9.96  256 

5 

33 

28 

9.60  012 

9.63  761 

0.36  239 

9.96  25l 

32 

29 

9.60  o4 1 

29 
29 

9.63  796 

35 
34 

0.36  2o4 

9.96  245 

S 

3i 

80 

9.60  070 

"dT 

9.63  83o 

d. 

0.36  170 

9.96  240 

30 

L.  Cos. 

L.  Cotg. 

L.  Tang. 

L.Sin.      d.j 

66°  30 . 

PP 

.1 

36 

35 

34 

.1 

30 

29 

.1 

6 

0.6 

5 

0.5 

3-6 

3-5 

M 

30 

2.0 

.2 

72 

7.0 

6.8 

.2 

6.0 

5.8 

.2 

1.2 

1.0 

•3 

10.8 

I 

o-S 

10.2 

3 

9.0 

8.7 

•3 

1.8 

I  5 

•4 

14.4 

I 

4.0 

•3.6 

•4 

I2.0 

11.6 

•4 

2.4 

2.0 

•5 

18.0 

1 

75 

17.0 

•  5 

150 

14.5 

•5 

30 

2-5 

.6 

21.6 

3 

I.O 

20.4 

.6 

18.0 

•7  4 

6 

3-6 

3.0 

•7 

25.2 

2 

4-5 

23.8 

•7 

21.0 

20.3 

•7 

4.2 

3-5 

.8 

28.8 

2 

8.0 

27.2 

,8 

24.0 

23.2 

.8 

4.8 

4.0 

-J 

._:2- 

27.0 

26  I 

76 


23°  30 

• 

' 

y 

L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

.  (io  070 

9.63  83o 

0.  36  I  70 

9.96  24o 

6 

30 

3i 

9.60  099 

29 

9. 63  863 

34 

0.36  i35 

9 

96  234 

29 

32 

9.60  128 

9.63  899 

0.36  loi 

9 

96  229 

6 

28 

33 

9.60  iSy 

9-63  934 

0.36  066 

9 

96  223 

27 

29 

34 

S 

34 

9.60  186 

29 

9.63  968 

35 

0.36  o32 

9 

96218 

6 

26 

35 

9.60  215 

9.64  oo3 

0.35  997 

9 

96  212 

25 

36 

9.60  244 

29 
29 

9.64  037 

34 
35 

0.35  963 

9 

96  207 

i 
6 

24 

37 

9.60  273 

9.64  072 

34 

0.35  928 

9 

96  201 

23 

38 

9  .60  302 

9.64  106 

0.35  894 

9 

96   196 

22 

39 

9.60  33i 

29 

9.64  i4o 

0.35  860 

9 

96   190 

21 

40 

9  .60  359 

9.64  175 

0.35  825 

9 

96185 

5 

6 

20 

4i 

9.60  388 

29 

9.64  209 

0.35  791 

9 

96   179 

19 

42 

9.60  417 

9.64  243 

0.35  757 

9 

96   174 

18 

43 

9.60  446 

29 
28 

9.64  278 

35 
34 

0.35  722 

9 

96  i68 

6 

17 

44 

9.60  474 

9.64  3l2 

34 

0.35  688 

9 

96  162 

16 

45 

9.60  5o3 

9-64  346 

0.35  654 

9 

96  157 

i5 

46 

9.60  532 

29 

9.64381 

35 

0.35  619 

9 

96  i5x 

i4 

47 

9.60  56i 

29 

9.64  415 

0.35  585 

9 

96  1 46 

6 

i3 

48 

9.60  589 

9.64  449 

0.35  551 

9 

96  i4o 

12 

49 

9.60  618 

29 

9.64  483 

34 

0.35  5i7 

9 

96135 

b 

II 

34 

50 

9.60  646 

29 

9.64  5i7 

35 

0.35 483 

9 

96  129 

6 

10 

6i 

9.60  675 

9.64  552 

0.35448 

9 

96   123 

9 

52 

9.60  704 

29 

9.64  586 

0.35  4i4 

9 

96  118 

8 

53 

9 .60  732 

28 

9.64  620 

34 

0.35  38o 

9 

96   112 

7 

54 

9.60  761 

29 

28 

9.64  654 

34 

0.35346 

9 

96   107 

5 

6 

55 

9.60  789 

9.64688 

0.35  3l2 

9 

96  lOI 

6 

5 

56 

9.60  818 

29 

9.64  722 

34 

0.35278 

9 

96  095 

4 

28 

34 

S 

57 

9.60846 

9.64  756 

0.35  244 

9 

96  090 

6 

3 

58 

9.60875 

29 

9.64  790 

0.35  210 

9 

96  o84 

2 

59 

9.60  903 

28 
28 

9.64824 

34 
34 

0.35  176 

9 

96079 

5 
6 

I 

60 

9.60  93 r 

9.64  858 

0.35  142 

9 

96  073 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.Tang. 

L.  Sin. 

d. 

' 

6<}°.                                                         1 

PP 

I 

35 

34 

— 

29 

38 

.1 

6 

5 

3-5 

3-4 

■  1 

2.9 

2.8 

0.6 

05 

.2 

7.0 

6.8 

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5-8 

5-6 

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1.2 

1.0 

•3 

10.5 

10.2 

3 

8.7 

8.4 

3 

1.8 

1-5 

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14.0 

13.6 

•4 

11.6 

II. 2 

•4 

2.4 

2.0 

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17  5 

17.0 

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14.5 

14.0 

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30 

2-5 

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21.0 

20.4 

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17.4 

16.8 

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3-6 

3.0 

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24.5 

23.8 

•7 

203 

19.6 

7 

4.2 

35 

.8 

28.0 

27.2 

.8 

23.2 

22.4 

.8 

4.8 

4.0 

— :a_ 

30.6 

.9    1      26.1 

^_ 

.^2- 

77 


24^ 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.60  931 

9.64  858 

0.35  142 

9.9G  073 

d 

60 

I 

9.60  960 

28 

9.64  892 

34 

0.35  108 

9.96  067 

s 

59 

2 

9.60988 

9.64  926 

0.35074 

9.96  062 

58 

3 

9.61  016 

29 

9.64  960 

34 
34 

0.35  o4o 

9.96  o56 

6 

i>7 

4 

9.61  045 

28 

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0.35  006 

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6 

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28 

9.65  062 

34 

0.34938 

9.96  039 

54 

7 

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29 
28 
28 

28 
28 

9.65  096 

34 

0.34  904 

9.96  o34 

6 

53 

8 

9.61  i58 

9  65  i3o 

0.34870 

9.96  028 

b2 

9 

9.61  186 

9.65  i64 

34 

33 

34 

0.34  836 

9.96  022 

5 
6 
6 

5i 

10 

9.61  2l4 

9.65  197 

0.34  8o3 

9.96  017 

50 

II 

9.61  242 

9.65  23i 

0.34  769 

9.96  01 1 

49 

12 

9.61  270 

9-65  265 

0.34  735 

9.96  oo5 

48 

i3 

9.61  298 

28 
28 

9.65  299 

34 

0.34  701 

9.96  000 

(, 

47 

i4 

9.61  326 

9.55333 

34 

0.34667 

9-95994 

6 

46 

i5 

9.61 354 

9.65  366 

33 

0.34  634 

9.95  988 

6 

45 

i6 

9.61  382 

28 

9-65  4oo 

34 

0.34  600 

9.95  982 

44 

'7 

9.61  4ii 

29 

9*.  65  434 

34 

0.34  566 

9.95977 

6 

43 

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9.61  438 

9.65467 

0.34533 

9.95  971 

42 

•9 

9.61  466 

28 
28 

9.65  5oi 

34 
34 

33 

0.34  499 

9.95  965 

S 
6 
6 

4i 
40 

20 

9.61  494 

9.65  535 

0.34  465 

9.95  960 

2  1 

9.61    522 

9.65  568 

0.34432 

9.95  954 

39 

22 

9.61  55o 

9.65  602 

34 

0.34  398 

9.95  948 

5 

38 

2  3 

9.61  578 

9.65  636 

34 

0.34  364 

9.95  942 

5 
6 

37 

24 

9.61  606 

9.65  669 

33 

0.34331 

9.95937 

36 

2b 

9.61  634 

9.65  703 

34 

0.34  297 

9.95  931 

35 

26 

9.61  662 

28 

9.65  736 

33 

0.34  264 

9.95  925 

34 

27 

9.61  689 

27 

9.65  770 

34 

0.34  23o 

9.95  920 

5 
6 

33 

28 

9.61  717 

9.65  8o3 

33 

0.34  197 

9.95  914 

g 

32 

29 

9.61  745 

28 

9.65  837 

34 
33 

0.34  i63 

9.95  908 

6 

3i 
30 

30 

9.61  773 

9.65  870 

0.34  i3o 

9.95  902 

L.  Cos. 

d. 

L.  Cotg.  !   d. 

L.  Tana-. 

L.  Sin.      d. 

' 

65°  30 . 

PP 

.1 

34 

33 

ag 

.1 

38 

37 

■  I 

6 

5 

3-4 

3-3 

2  9 

t.8 

2.7 

0.6 

0-5 

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6.8 

6.6 

5.8 

2 

,.6 

54 

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1.2 

1.0 

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10.2 

9.9 

8-7 

•3 

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i.4 

8.1 

•3 

1.8 

1-5 

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13.2 

11.6  ; 

■4 

I 

.2 

laS 

•4 

2-4 

2.0 

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17.0 

Ib.s 

MS 

•5 

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.0 

'3-5 

•5 

30 

2S 

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20.4 

19.8 

17-4 

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i( 

).8 

16.2 

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3-6 

3.0 

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238 

23 1 

20.3 

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K 

>6 

18.0 

7 

42 

3S 

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27.2 

26.4 

23.2 

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2= 

'4 

21.6 

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4.8 

4.0 

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2^^__ 

26.1 

2 

.2 

78 


24 

°30 

• 

/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.      d.  1 

30 

y.bi  773 

27 
28 

9.66  870 

0.34  1  3o 

9.95  902 

30 

3i 

9.61  800 

9-65  904 

33 

0.34  096 

9.95897 

6 

29 

32 

9.61  828 

28 

9-65  937 

34 

0.34  o63 

9. 95  891 

5 

28 

33 

9.61  856 

9-65  971 

0.34  029 

9.95885 

6 

27 

34 

9.6t  883 

28 

9.66  oo4 

34 

0.33  996 

9.95  879 

6 

26 

35 

9.61  911 

28 

9.66  o38 

33 

0.33  962 

9.95  873 

25 

36 

9.61  939 

9.66  071 

0.33  929 

9.95  868 

24 

27 

33 

'^7 

9.61  966 

28 

9.66  io4 

0.33896 

9.95  862 

6 

23 

3« 

9.61  994 

9.66  i38 

0.33  862 

9.95  856 

6 
6 

5 
6 

22 

39 

9.62  021 

27 

28 

27 
28 

9.66  171 

33 
33 

34 

0.33  829 

9.95  85o 

21 

20 

'9 

40 

9.62  049 

9.66  2o4 

0.33  796 

9.95844 

4i 

9.62  076 

9.66  238 

0.33  762 

9.95  839 

42 

9.62  io4 

9.66  271 

0.33  729 

9.95  833 

18 

43 

9.62  i3i 

28 

9.66  3o4 

33 

0.33  696 

9.95  827 

6 

17 

44 

9.62  159 

27 

9.66  337 

34 

0.33  663 

9.95  821 

fi 

16 

45 

9.62  186 

9.66  371 

0.33  629 

9.95  8i5 

i5 

46 

9.62  2l4 

27 

9.66  4o4 

33 
33 

0.33  596 

9.95  810 

6 

i4 

47 

9.62  24l 

9.66  437 

0.33  563 

9.95  8o4 

6 

i3 

48 

9.62  268 

9.66  470 

0.33  53o 

9.95  798 

6 

12 

49 

9.62  296 

27 
27 

9.66  5o3 

33 
34 
33 

0.33  497 

9.95  792 

6 
6 

1 1 
10 

9 

50 

9.62  323 

9.66  537 

0.33  463 

9.95  786 

5i 

9.62  35o 

9.66  570 

0.33  43o 

9.95  780 

52 

9.62  377 

28 

9.66  6o3 

0.33  397 

9-95  775 

fi 

8 

53 

9.62  405 

9.66  636 

33 

0.33  364 

9.95  769 

6 

7 

27 

33 

54 

g.62  432 

27 

9.66  669 

0.33  33i 

9.95  763 

6 

6 

55 

9.62  459 

9.66  702 

0.33  298 

9.95  757 

6 

5 

56 

9.62486 

27 

9.66  735 

0.33  265 

9.95  75i 

6 

4 

27 

33 

^7 

9.62  5i3 

28 

9,66768 

33 

0.33  232 

9.96  745 

6 

3 

58 

9.62  54i 

9.66801 

0.33  199 

9.95  739 

6 

2 

59 

9.62  568 

27 

9.66  834 

33 

0.33  166 

9.95  733 

I 

60 

9.62  595 

27 

9.66  867 

33 

0.33  i33 

9.95  728 

0 

L.  Cos. 

1  d. 

L.  Cotg.  I  d. 

L.  Tang. 

L.  Sin. 

d. 

65°.                                                         1 

PP 

I 

34 

33 

28 

37 

.1 

6 

5 

3-4 

3-3 

.1 

2.8 

2.7 

0.6 

o-S 

.2 

6.8 

6.6 

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5-6 

5-4 

.3 

1.2 

I.O 

•3 

10.2 

9  9 

3 

8.4 

8.1 

•3 

1.8 

1-5 

■4 

13-6 

13.2 

■4 

II. 2 

10.8 

•4 

2.4 

2.0 

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17.0 

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5 

14.0 

»3-5 

•5 

3.0 

25 

.6 

20.4 

19.8 

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16.8 

16.2 

6 

3.6 

30 

•7 

23.8 

23- 1 

•7 

19.6 

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.7 

4.2 

3-5 

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27.2 

26.4 

8 

22.4 

21.6 

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4.8 

4.0 

30.6 

9           25.2 

.^ 

•9             5-4 

79 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.62  595 

9.^6867 

0.33  i33 

9.95  728 

6 

60 

I 

9.62  622 

27 

9.66  900 

33 

0.33  100 

9.95  722 

6 

59 

2 

9.62  649 

9.66  933 

0.33  067 

9.95  716 

58 

3 

9.62  676 

27 

9.66  966 

33 

0.33  o34 

9.95  710 

6 

57 

4 

9,62  708 

27 

9.66999 

33 

0.33  001 

9.95  704 

6 

56 

6 

9.62  780 

9.67  082 

0.32968 

9.95  698 

55 

6 

9.62  767 

27 

9.67065 

33 

0.32  935 

9.95  692 

6 

34 

7 

9.62  784 

9.67  ogS 

33 

0.82  902 

9.95  686 

6 

53 

8 

9.62  81 1 

9.67  i3i 

0.32  869 

9.95  680 

52 

9 

9.62838 

9.67  i63 

0.82887 

9.95  674 

5i 

27 

10 

9.62  865 

9.67  196 

0.82  8o4 

9.95  668 

50 

1 1 

9.62  892 

26 

9.67  229 

33 

0.82  771 

9.95  663 

6 

49 

12 

9.62  918 

9.67  262 

0.82  788 

9.95  657 

48 

i3 

9.62  945 

27 

9.67295 

32 

0.82  7o5 

9.95  65i 

6 

47 

i4 

9.62  972 

27 

9.67327 

33 

0.82  678 

9.95645 

6 

46 

i5 

9.62999 

9.67  36o 

0.82  64o 

9.95  689 

45 

i6 

9.63  026 

26 

9.67393 

33 

0.82  607 

9.95  633 

6 

44 

'7 

9.63  o52 

27 

9.67  426 

32 

0.82  574 

9.95  627 

6 

48 

i8 

9.63  079 

9.67  458 

0.82  542 

9.95  621 

6 

42 

'9 

9.63  106 

9.67  491 

0.82  509 

9.95  615 

4i 

20 

9.63  i33 

9.67  624 

0.82  476 

9.95  609 

40 

21 

9.63  159 

9.67556 

0.82  444 

9.95  608 

6 

39 

22 

9.63  186 

9.67  589 

0.82  4t  I 

9.95  597 

6 

38 

23 

9.63  2l3 

9.67  622 

0.82  878 

9.95  591 

37 

26 

32 

6 

24 

9.63  239 

27 
26 

9.67  654 

0.82  846 

9.95585 

6 

86 

25 

9.63  266 

9.67687 

0.32  818 

9.95579 

6 

85 

26 

9.63  292 

9.67719 

32 

0.82  281 

9.95  578 

34 

27 

33 

b 

27 

9-63  319 

26 

9.67  752 

0.82  248 

9.95  567 

f, 

38 

2  8 

9.63345 

9.67785 

0.82  2l5 

9.95  56i 

32 

29 

9.63  372 

27 

9.67  817 

32 

0.82  i83 

9.95  555 

6 

3i 

30 

9.63  398 

9.67  850 

33 

0.82  I  5o 

9.95  549 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

64°  30 . 

1 

PP 

.1 

33 

3a 

ay 

36 

.1 

6 

5 

3  s 

3-2 

.1 

2.7 

2.6 

0.6 

05 

.2 

6.6 

6.4 

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5-4 

5-2 

.3 

1.3 

1.0 

•3 

9.9 

9.6 

•3 

8.1 

7.8 

•3 

1.8 

1-5 

•4 

13.2 

12.8 

•4 

10.8 

10.4 

•4 

2-4 

3.0 

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16. 5 

16.0 

•5 

13-5 

13.0 

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30 

2-5 

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19.8 

19.2 

6 

16.2 

15.6 

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3-6 

30 

•7 

a3i 

22.4 

•7 

18.9 

t8.2 

■7 

43 

35 

.8 

26.4 

25.6 

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21.0 

20.8 

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4.8 

4.0 

1             -9           29-7       i 

28.8 

^^ 

80 


2 

•0 

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'. 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.63  398 

27 

9.67  850 

32 

0.32  i5o 

9.95  549 

6 

30 

3i 

9-63  425 

26 

9.67  882 

0.32  118 

9.95  543 

g 

29 

32 

9.63  451 

9.67915 

0.32  o85 

9.95  537 

28 

33 

9.63478 

27 
26 

9.67947 

32 

33 

0.32  o53 

9.95  53i 

6 
6 

27 

34 

9.63  5o4 

27 

9.67  980 

0.32  020 

9.95  525 

6 

26 

35 

9.63  53i 

9.68  012 

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9.95  5i9 

25 

36 

9.63  557 

26 

9.68044 

32 

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9.95  5i3 

6 

24 

37 

9.63  583 

27 

9.68  077 

33 
32 

o.3i  923 

9.95  5o7 

23 

38 

9.63  610 

9.68  109 

o.3i  891 

9.95  5oo 

22 

39 

9.63  636 

9.68  142 

33 

o.3i  858 

9.95  494 

6 
6 

21 

40 

9-63  662 

9.68  174 

32 

o.3i  826 

9.95  488 

20 

4i 

9.63689 

26 

9.68  206 

33 

o.3i  794 

9.95  482 

19 

42 

9.63  715 

26 

9.68  239 

o.3i  761 

9.95  476 

6 
6 

18 

43 

9-63  741 

26 

9.68  271 

32 

o.3i  729 

9.95  470 

17 

44 

9.63  767 

27 

9.68  3o3 

33 

o.3i  697 

9.95  464 

6 

16 

45 

9-63  794 

9.68  336 

0.3 1  664 

9.95458 

6 
6 
6 

i5 

46 

9.63  820 

9.68  368 

32 

o.3i  632 

9.95  452 

i4 

47 

9.63  846 

26 

9.68  4oo 

32 

o.3r  600 

9.95  446 

i3 

48 

9.63  872 

26 

9.68432 

o.3i  568 

9.95  44o 

6 

12 

49 

9.63  898 

9.68  465 

33 

o.3i  535 

9.95434 

1 1 

50 

9.63  924 

26 
26 

9.68  497 

32 
32 

o.3i  5o3 

9.95  427 

1 
6 
6 

10 

9 

5i 

9.63  950 

9.68  529 

o.3i  471 

9.95  421 

52 

9.63  976 

9.68  56i 

o.3i  439 

9.95  4i5 

6 

8 

53 

9.64  002 

9.68593 

32 

o.3i  407 

9.95  409 

7 

26 

33 

6 

i>4 

9.64  028 

26 

9.68626 

32 

o.3i  374 

9.95  4o3 

6 

6 

55 

9.64  o54 

26 

9.68  658 

o.3i  342 

9.95  397 

6 

5 

56 

9.64  080 

9.68  690 

32 

o.3i  3io 

9.95  391 

4 

26 

32 

7 

i)7 

9.64  106 

26 

g.68  722 

32 

o.3i  278 

9.95  384 

6 

3 

58 

9.64  l32 

9.68  754 

o.3i  246 

9,95  378 

6 
6 

2 

59 

9.64  i58 

26 

9.68  786 

32 

32 

o.3i  2i4 

9.95  372 

I 
0 

60 

9.64  i84 

9.68  818 

o.3i  182 

9.95  366 

L.  Cos. 

"d. 

L.  Cotg. 

d. 

L,  Tang. 

L.  Sin.      d. 

' 

64°.                                                         1 

PP  1       33 

3a 

47 

36 

7 

6 

.1 

3-3 

3-2 

.1 

2-7 

2.6 

., 

0.7 

0.6 

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6.6 

6.4 

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5-4 

5-2 

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1.2 

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9-9 

9.6 

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8.1 

7.8 

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2.1 

1.8 

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13.2 

12.8 

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10.8 

104 

•4 

2.8 

2-4 

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16.  s 

16.0 

■5 

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13.0 

5 

3-S 

3.0 

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19.8 

19.2 

.6 

16.2 

15.6 

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4-2 

3.6 

•7 

23- « 

22.4 

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18.2 

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4-9 

4-2 

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26.4 

25.6 

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21.6 

20.8 

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5.6 

4.8 

29.7 

28.8        1                  .Q      '          24.3 

■Q     1         6.3        1         .S.4       1 

81 


26°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.64  184 

.26 
26 

9.68818 

32 
32 

o.3i  182 

9.95  366 

6 
6 

60 

I 

9.64  210 

9.68  85o 

o.3i  150 

9.95  36o 

59 

2 

9.64  236 

26 

9.68882 

o.3i  118 

9.95354 

5 

58 

3 

9.64  262 

26 

9.68  914 

32 

o.3i  086 

9.95  348 

^7 

4 

9.64288 

25 

9.68  946 

o.3i  o54 

9.95  341 

6 

56 

5 

9.643i3 

26 

9.68978 

o.3i  022 

9.95  335 

5 

55 

6 

9.64  339 

26 

9.69  010 

32 

o.3o  990 

9.95  329 

6 

54 

7 

9.64  365 

26 

9.69  042 

o.3o  958 

9.95  323 

6 

53 

8 

9.64  391 

26 

9.69  074 

o.3o  926 

9.95317 

52 

9 

9.64  417 

25 
26 
26 

9.69  106 

32 
32 

32 

o.3o  894 

9.95  3io 

6 

6 
6 

5i 

10 

9.64  442 

9.69  i38 

o.3o  862 

9.95  3o4 

50 

II 

9.64468 

9.69  170 

o.3o83o 

9.95  298 

49 

12 

9.64494 

9.69  202 

o.3o  798 

9.95  292 

48 

i3 

9 .  64  5 1 9 

26 

9.69  234 

32 

o.3o  766 

9.95  286 

47 

i4 

9.64545 

26 

9.69  266 

32 

o.3o  734 

9.95279 

6 

46 

i5 

9.64  571 

9.69  298 

o.3o  702 

9.95  273 

6 
6 

45 

i6 

9.64  596 

25 

26 

9.69  329 

31 

o.3o  671 

9.95  267 

44 

17 

9.64  622 

9.69  36i 

o.3o  639 

9.95  261 

7 

6 
6 

6 

43 

i8 

9.64  647 

9.69  393 

o.3o  607 

9.95  254 

42 

'9 

9.64  673 

25 

9.69425 

32 
32 

o.3o  575 

9.95  248 

4i 

20 

9.64698 

9.69  457 

o.3o  543 

9.95  242' 

40 

21 

9.64  724 

23 

9.69488 

o.3o  5i2 

9.95  236 

39 

22 

9.64  749 

9.69  520 

o.3o48o 

9.95  229 

38 

23 

9-64  775 

25 

9.69  552 

32 
32 

o.3o448 

9.95  223 

6 

37 

24 

9.64  800 

26 

9.69  584 

31 

o.3o  4i6 

9.95  217 

6 

36 

25 

9.64826 

9.69  6i5 

o.3o385 

9.95  211 

35 

26 

9.64  851 

25 
26 

9.69  647 

32 
32 

o.3o  353 

9.95  204 

7 
6 

34 

27 

9.64877 

9.69679 

o.3o  321 

9.95   198 

6 

33 

28 

9.64  902 

9.69  710 

o.3o  290 

9.95   192 

32 

29 

9.64  927 

25 
26 

9.69  742 

32 

0.30  258 

9.95  i85 

7 

6 

3i 

30 

9.64  953 

9.69774 

32 

o.3o  226 

9.95  179 

30 

L.  Cos. 

d. 

L.  Cotg.  1   d. 

L.  Tang. 

L.  Sin. 

d. 

' 

63°  30 

'. 

PP 

.1 

3a 

31 

36 

as 

.1 

7 

6 

l" 

3-' 

.1 

2.6 

2-5 

07 

0.6 

.2 

6.4 

6.2 

.2 

5-2 

50 

.2 

1-4 

1.2 

•3 

9.6 

9-3 

•3 

7.8 

7-5 

•3 

2.1 

1.8 

•4 

12.8 

12.4 

•4 

10.4 

lO.O 

•4 

2.8 

2-4 

■5 

16.0 

'5-5 

•5 

13.0 

12.5 

•5 

3-5 

30 

.6 

19.2 

18.6 

.6 

15.6 

150 

.6 

4-2 

3-6 

•7 

22.4 

21.7 

•7 

18.2 

17s 

.7 

4  9 

42 

.8 

256 

24.8 

.8 

2a  8 

20.0 

.8 

5.6 

4.8 

.Q               28.8         1 

^^ 

•  9           f<-i 

82 


26° 

30. 

/ 

L.Sin.    !  d. 

L.  Tang,  j  d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.64  953 

25 

9.69  774 

3' 
32 

o.3o  226 

9.95  179 

6 
6 

30 

29 

3i 

9.64  978 

9.69  8o5 

o.3o  195 

9-95  173 

32 

9.65  oo3 

9.69  837 

o.3o  i63 

9.95  167 

28 

33 

9.65  029 

9.69868 

31 

o.3o  i32 

9.95  i6o 

7 

27 

25 

'    32 

34 

9-65  o54 

9.69  900 

o.3o  100 

9.95  1 54 

6 

26 

35 

9-65  079 

9.69  932 

o.3o  068 

9.95  i48 

25 

36 

9-65  io4 

25 
26 

9.69963 

31 

32 

o.3o  037 

9.95  i4i 

6 

24 

37 

9.65  i3q_ 

9.69995 

31 

o.3o  oo5 

9.95  135 

6 

23 

3S 

9.65  155 

9.70  026 

0.29  974 

9.95  129 

22 

39 

9.65  180 

25 

25 
25 

9.70  o58 

32 
31 

32 
31 

0.29  942 

9.95  122 

6 

6 

7 

21 

40 

9.65  205 

9.70  089 

0.29  91 1 

9.95  116 

20 

4i 

9-65  23o 

9.70  121 

0.29  879' 

9.95  no 

19 

42 

9.55  255 

9.70  l52 

0.29  848 

9.95  io3 

18 

43 

9-65  281 

25 

9.70  184 

32 
31 

0.29  816 

9.95097 

7 

17 

44 

9.65  3o6 

9.70  2l5 

32 

0.29  785 

9.95  090 

6 

16 

45 

9.65  33i 

9.70247 

0.29  753 

9.95  084 

6 

i5 

46 

9.65  356 

25 

25 

9.70  278 

3' 
31 

0.29  722 

9.95  078 

7 

i4 

47 

9.65  38i 

9.70  309 

32 

0.29  691 

9.95  071 

6 

i3 

48 

9.65  4o6 

9.70  34i 

0.29  659 

9.95  065 

5 

12 

49 

9.65431 

25 

9.70  372 

3' 

0.29  628 

9.95  059 

7 
6 

II 

50 

9.65  456 

25 

9.70  4o4 

32 

0.29  596 

9.95  o52 

10 

25 

3' 

5i 

9.65481 

9.70435 

0.29  565 

9.95  o46 

7 

9 

52 

9.65  5o6 

9.70  466 

0.29  534 

9.95  039 

g 

8 

53 

9.65  53i 

25 

9.70498 

32 

0.29  5o2 

9.95  o33 

6 

7 

54 

9.65  556 

g.70  529 

31 

0.29  471 

9.95  027 

7 

6 

55 

9.65  58o 

9.70  56o 

0.29  44o 

9.95  020 

g 

5 

56 

9.65  6o5 

25 

9.70  592 

32 

0.29  4o8 

9.95  oi4 

4 

5? 

9.65  63o 

25 

25 

9.70  623 

31 

0.29  377 

9.95  007 

6 

3 

58 

9.65  655 

9.70  654 

0.29  346 

9.95  001 

6 

2 

59 

9.65  680 

25 

9.70  685 

31 

0.29  315 

9-94995 

7 

1 

60 

9.65  705 

25 

9.70  717 

32 

0.29  283 

9.94988 

0 

L.  Cos. 

d. 

L.  Cotg.  1   d. 

L.  Tang. 

L.  Sin. 

d. 

f 

63°.                                                       1 

PP 

.1 

33 

31 

26 

.1 

as 

34 

.1 

7 

6 

3-2 

31 

2.6 

!-5 

2.4 

0.7 

0.6 

.2 

6.4 

6.2 

5-2 

2 

;.o 

4.8 

.2 

1.4 

1.2 

•3 

9.6 

9-3 

7.8 

•3 

r-5 

7.2 

•3 

2.1 

1.8 

•4 

12.8 

12.4 

10.4 

•4 

i( 

5.0 

9,6 

•4 

2.8 

2.4 

•S 

16.0 

15- S 

13.0 

•5 

I 

2-5 

12.0 

5 

3-5 

30 

.6 

19.2 

18.6 

15-6 

.6 

I 

j.O 

14.4 

.6 

4.2 

3-6 

•7 

22.4 

21.7 

18.2 

•7 

I 

^5 

16.8 

•7 

4-9 

4.2 

.8 

25.6 

24.8 

20.8 

8 

2( 

).o 

19.2 

.8 

5-6 

4.8 

28.8    1    27.9    1 

2 

'J- 

21.6 

6-3             5-4     1 

83 


27' 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

^^'~ 

0 

9.65  705 

24 

25 

9.70717 

31 
31 

0.29283 

9.94988 

6 

7 
6 

60 

59 

I 

9.65  729 

9.70  748 

0.29  252 

9.94  982 

2 

9.65  754 

25 

9.70779 

0.29  221 

9.94975 

58 

3 

9.65779 

9.70  810 

31 
32 

0.29  190 

9.94969 

57 

4 

9.65  8o4 

24 

9.70  84i 

0.29  I  59 

9.94962 

7 

6 

56 

b 

9.65  828 

9.70873 

0.29  127 

9.94956 

55 

6 

9.65  853 

9.70  904 

31 

0.29  096 

9-94949 

7 

54 

7 

9.65  878 

24 

9.70935 

0.29  o65 

9.94943 

7 
6 

7 

6 

6 

53 

8 

9.65  902 

9.70  966 

0.29034 

9.94936 

52 

9 

9.65927 

25 

24 

25 

9.70997 

31 

3« 
3» 

0.29  oo3 

9.94  930 

5i 

10 

9.65  952 

9.71  028 

0.28  972 

9.94  923 

50 

49 

1 1 

9.65  976 

9-7 

I  059 

0.28  94 1 

9.94917 

12 

9.66  001 

9.7 

r  090 

0,28  910 

9.94911 

48 

i3 

9.66  025 

25 

9-7 

[  121 

32 

0.28  879 

9.94904 

7 
5 

47 

i4 

9.66  o5o 

25 

9-7 

r  i53 

31 

0.28  847 

9.94898 

46 

i5 

9.66075 

9-7 

184 

3» 

0.28  816 

9.94891 

6 

45 

i6 

9.66  099 

25 

9-7 

215 

0,28  785 

9.94885 

44 

n 

9.66  124 

24 

9.71 

246 

3' 

0.28  754 

9.94878 

43 

i8 

9.66  i48 

25 

24 

9-7 

277 

0.28  723 

9.94871 

42 

•9 

9.66  173 

9.71 

3o8 

31 

0.28  692 

9.94865 

7 

4i 

20 

9.66  197 

9.71 

339 

0.28661 

9.94  858 

40 

21 

9.66  221 

25 

9.71 

370 

3' 

0.28  63o 

9.94  852 

39 

22 

9.66  246 

9.71 

4oi 

30 

0.28  599 

9.94845 

38 

23 

9.66  270 

25 

24 

9.71 

43i 

0.28  569 

9.94839 

37 

24 

9.66295 

9.71 

462 

31 

0.28  538 

9.94832 

7 
6 

36 

25 

9.66  319 

9.71 

49^ 

0.28  5o7 

9.94826 

35 

26 

9.66343 

25 

9.71 

624 

3' 
31 

0.28  476 

9.94  819 

7 
6 

34 

27 

9.66  368 

9.71 

555 

0.28  445 

9.94813 

33 

28 

9.66  392 

9.71 

586 

0.28414 

9.94  806 

32 

29 

9.66416 

24 

9.71 

617 

0.28  383 

9.94799 

7 

3i 

30 

9.66  44i 

9.71  648 

0.28  352 

9.94793 

30 

L.  Cos.      d.  1  L.Cotg.  1  d.  | 

L.  Tang. 

L.Sin.      d.| 

' 

62°  30 

1 

PP 

.  I 

3a 

31 

30 

.1 

as 

24 

.  I 

7 

6 

3-2 

31 

30 

2 

5 

2.4 

0.7 

0.6 

.2 

6.4 

6.2 

6.0 

.2 

5 

0 

4.8 

.2 

»-4 

1.2 

•3 

9.6 

93 

9.0 

•3 

7 

5 

7.2 

•3 

2.1 

1.8 

•4 

12.8 

124 

12.0 

■4 

10 

0 

9.6 

.4 

2.8 

2.4 

•5 

16.0 

'55 

15.0 

•  5 

12 

5 

I2.0 

•5 

3-5 

30 

.6 

19.2 

18.6 

18.0 

.6 

15 

0 

14.4 

.6 

4-2 

3-6 

•7 

22.4 

21.7 

21.0 

•7 

17 

5 

168 

•7 

4-9 

4.2 

.8 

256 

24.8 

24.0 

.8 

20 

0 

ig.2 

.8 

5-6 

4.8 

g    1      28.8     1      27. q     1    27.0 

9    \     22 

5      21.6    1 

6.^      1      ..4     1 

84 


27 

0 

30 

'• 

' 

L.  Sin. 

d. 

L.  Tang.  !  d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9,66  44i 

9.71  648 

0.28  352 

9.94793 

30 

3t 

9.66465 

9.71  679 

30 

0.28  321 

9.94  786 

6 

29 

32 

9.66489 

9.71  709 

0.28  291 

9.94  780 

28 

33 

9.66  5i3 

24 

9.71  740 

0.28  260 

9.94773 

6 

27 

34 

9.66537 

9.71  771 

0.28  229 

9.94767 

26 

35 

9.66  562 

9.71  802 

0.28  198 

9.94  760 

25 

36 

9.66  586 

24 

9.71  833 

30 

0.28  167 

9-94753 

6 

24 

37 

9.66  610 

9.71  863 

0.28  137 

9.94747 

23 

38 

9.66634 

9.71  894 

0.28  106 

9.94  740 

6 

22 

39 

9.66  658 

24 

9.71  925 

31 

0.28  075 

9.94  734 

21 

40 

9,66  682 

24 

9.71  955 

31 

0.28  045 

9.94727 

7 

20 

4i 

9.66  706 

25 

9.71  986 

0.28  oi4 

9.94  720 

6 

19 

42 

9.66  73i 

9.72  017 

0.27983 

9.94  7^4 

18 

43 

9.66  755 

24 
24 

9.72  o48 

31 

30 

0.27  952 

9.94707 

1 
7 

17 

44 

9.66779 

9.72  078 

0.27  922 

9.94  700 

fi 

16 

45 

9.66  8o3 

9.72  109 

0.27  891 

9.94  694 

i5 

46 

9.66  827 

24 

24 

9.72  i4o 

31 
30 

0.27  860 

9.94  687 

7 

i4 

47 

9.66  851 

9.72  170 

3' 

0.27  83o 

9.94  680 

6 

i3 

48 

9.66  875 

9.72  201 

0.27799 

9.94  674 

12 

49 

9.66  899 

24 
23 

24 

9.72  23l 

30 
31 

31 

0.27  769 

9.94667 

7 
7 
6 

7 

1 1 

10 

9 

50 

9.66  922 

9.72  262 

0.27  738 

9.94  660 

5i 

9.66  946 

9.  72  293 

0.27  707 

9.94  654 

52 

9.66  970 

9.72  323 

0.27  677 

9.94647 

8 

53 

9.66994 

24 

9.72  354 

31 

0.27  646 

9.94  64o 

7 

54 

9.67  018 

24 

9.72  384 

30 

0.27  616 

9.94634 

7 

6 

55 

9.67  042 

9.72415 

0.27  585 

9.94627 

5 

56 

9.67  066 

24 

9.72445 

30 

0.27  555 

9.94  620 

7 
6 

4 

57 

9.67  090 

24 

9.72476 

31 

0.27  524 

9.94  6i4 

3 

58 

9.67  1 13 

9. 72  5o6 

0.27494 

9.94607 

2 

59 

9.67  i37 

24 
24 

9.72  537 

31 

30 

0.27  463 

9.94  600 

7 

I 
0 

60 

9.67  161 

9. 72  567 

0.27  433 

9.94  593 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

62°.                                                        1 

PP 

.1 

31 

30 

25 

.1 

24 

23 

.t 

7 

6 

31 

30 

2-5 

2.4 

2-3 

0.7 

0.6 

.2 

6.2 

6.0 

50 

.2 

4.8 

4.6 

.2 

1.4 

1.2 

■3 

9-3 

9.0 

7-5 

•3 

7.2 

6.9 

•3 

2.1 

•4 

12.4 

12.0 

10. 0 

•4 

9,6 

9.2 

■4 

2.8 

2.4 

•  5 

15-5 

15.0 

12.5 

•  5 

12.0 

II-5 

•s 

3-5 

30 

.6 

18.6 

18.0 

15.0 

.6 

14.4 

13.8 

.6 

4.2 

3.6 

•  7 

21.7 

21.0 

17- 5 

•7 

16.8 

16. 1 

•7 

4<3 

4.2 

.8 

24.8 

24.0 

20.0 

.8 

19.2 

18.4 

.8 

5-b 

4.8 

27.9 

27.0 

22.5 

.9     1      2r.6     1      20.7     1 

•q     i       6.: 

_        5-A_J 

85 


4 
5 
6 

7 

8 

9 
10 

1 1 

12 

i3 

i4 
i5 
i6 

'7 
i8 

'9 

20 


22 
23 

24 
25 

26 

27 
28 
29 

30 


L.  Sin. 

9.67  ibi 


.67  185 
.67  208 

.67  232 

.67  256 
.67  280 
,67  3o3 

g.67  327 
9.67  35o 
9.67  374 


9.67  398 


9.67  421, 
9.67445 
9.67468 

9.67  492 
9.67  5i5 
9.67  539 

9.67  562 
9.67  586 
9.67  609 


9.67  633 


9.67  656 
9.67  680 
9.67  703 

9.67  726 
9.67  750 
9.67  773 

9.67796 
9.67  820 
9.67  843 

9.67  866 

L.  Cos. 


d. 

24 
23 
24 
24 
24 
23 
24 
23 
24 
24 
23 
24 
23 
24 
23 
24 
23 
24 
23 
24 

23 
24 
23 
23 
24 
23 
23 
24 
23 
23 

T" 


L.  Tang. 

9.72  567 


72  598 
72  628 
72  659 

72  689 
72  720 
72  750 

72  780 
72  81 1 
72  84i 


9.72  872 


28°. 
~ 

31 
30 

31 
30 
31 
30 
30 

31 

30 

31 


,72  902 
.72  932 
,72  963 

,72993 
,73  023 
73  o54 

,73084 
,73  ii4 
73  i44 


9.73  175 


,73  205 
,73235 
,73  265 

,73  295 
.73  326 
,73356 

,73386 
,73416 
,73446 


9.73  476 
L.  Cotg. 


L.  Cotg. 

0.27  433 


.27  4o2 
.27  372 
.27  341 

,27  3i  I 
.27  280 
.27  250 

.27  220 
.27  189 
,27  159 


0.27  128 


,27  098 
,27  068 
,27  037 

,27  007 
,26  977 
,26  946 

.26  916 

.26886 
.26  856 


0.26  825 


0.26  795 
0.26  765 
0.26  735 

0.26  705 
0.26  674 
0.26  644 

0.26  6i4 
0.26  584 
0,26  554 

0.26  524 


d.  I  L.  Tang. 


L.  Cos. 

9.94  593 

94587 
94  58o 
94  573 

94  567 
94  56o 
94553 

94546 
94  540 
94  533 


94  526 


94  519 
94  5i3 
94  5o6 

94499 
94  492 
94  485 

94  479 
94  472 
94  465 


94  458 


94  45 1 
94  445 
94  438 

94431 
94  424 
94417 

94  4io 
94  4o4 
94  397 

94  390 

L.  Sin.    ^ 


61° 30. 


PP 


31 
6.2 


12.4 

J5-5 
18.6 

21.7 
24.8 
27  9 


3.0 
6.0 


12.0 
15.0 
18.0 

21.0 
24.0 
27.0 


»4 

2.4 
4.8 
7.2 

9.6 

I2.0 
14.4 


86 


2.3 
4.6 
6.9 

9.2 

J'S 

138 

16. 1 

18.4 
20.7 


0.7 
1.4 
2.1 

2.8 

3-5 
4-2 

4.9 
5-6 
6-3 


28 

°3C 

y. 

L.  Sin.    1   d.  1 

L.  Tang. 

d.      L.  Cotg.  1 

L.  Cos.      d. 

30 

9.67866 

9.73  476 

0.26  524 

9.94  390 

30 

3i 

9- 

67  890 

23 

9.73  507 

30 

0.26  493 

9.94383 

29 

32 

9- 

67  913 

9.73537 

0.26  463 

9-94  376 

28 

33 

9- 

67  936 

23 
23 

9.73  567 

30 

0.26433 

9.94  369 

27 

34 

9 

67959 

23 

9.73597 

30 

0.26  4o3 

9.94  362 

26 

35 

9 

67  982 

9.73  627 

0.26  373 

9.94355 

25 

36 

9 

68006 

23 

9.73  657 

30 

0.26343 

9.94  349 

24 

37 

9 

68  029 

9.73  687 

30 

0.26  3i3 

9.94342 

23 

38 

9 

68o52 

9.73717 

0.26283 

9.94335 

22 

39 

9 

68075 

23 

9.73747 

0.26253 

9.94  328 

21 

40 

9 

68  098 

9.73777 

0.26  223 

9.94  321 

20 

4i 

9 

68  121 

23 

9.73  807 

0.26  193 

9.94  3i4 

19 

42 

9 

68  i44 

9.73337 

0.26  i63 

9.94  3o7 

18 

43 

9 

68  167 

23 

9.73  867 

0.26  i33 

9.94  3oo 

17 

44 

9 

68  190 

23 

9.73897 

0.26  io3 

9.94  293 

16 

45 

9 

68213 

9.73927 

0.26  073 

9.94  286 

i5 

46 

9 

68237 

24 
23 

9.73957 

30 
30 

0.26  043 

9.94279 

i4 

47 

9 

68  260 

9.73987 

0.26  oi3 

9.94  273 

i3 

48 

9 

68283 

9.74017 

0.25  983 

9.94  266 

12 

49 

9 

68  3o5 

22 

9.74047 

30 

0.25  953 

9.94  259 

II 

50 

9 

68  328 

23 
23 

9.74077 

30 
30 

0.25  923 

9.94  252 

10 

9 

5i 

9 

68  35i 

9.74  107 

0.25  893 

9.94  245 

52 

9 

68374 

23 

9.74  137 

0. 25  863 

9.94238 

8 

53 

9 

68397 

23 

9.74  166 

29 

0. 25  834 

9.94  23l 

7 

54 

9 

68420 

23 

9.74  196 

30 

0.25  8o4 

9.94  224 

6 

55 

9 

.68  443 

9.74  226 

0.25  774 

9.94  217 

5 

56 

9 

.68  466 

23 

9.74  256 

30 

0.25  744 

9.94  210 

4 

57 

9 

.68489 

23 

9.74  286 

30 

0.25  714 

9.94  2o3 

3 

58 

9 

.68  512 

23 

9.74  3i 6 

0.25  684 

9-94  196 

2 

59 

9 

.68  534 

22 
23 

9-74345 

29 
30 

0.25  655 

9.94  189 

I 
0 

60 

9 

.68  557 

9-74375 

0.25  625 

9.94  182 

L.  Cos. 

d. 

L.  Cotg. 

d.  1  L.  Tang.  | 

L.Sin.    Id. 

' 

61°.                                                         1 

PP 

3t 

30 

29 

.1 

24 

23 

23 

.1 

7 

6 

31 

30 

2-9 

2.4 

2-3 

2.2 

0.7 

06 

.2 

6.2 

6.0 

5-8 

.2 

4.8 

4.6 

4.4 

.2 

1.4 

1.2 

■3 

9-3 

9.0 

8.7 

■3 

7.2 

6.9 

6.6 

•3 

2.1 

1.8 

•4 

12.4 

12.0 

11.6 

■4 

9.6 

9.2 

8.8 

•4 

2.8 

2.4 

•5 

iSS 

15.0 

14s 

•  s 

12.0 

"•5 

II. 0 

•  5 

3-5 

3.0 

.6 

18.6 

18.0 

'7-4 

.6 

14.4 

13.8 

13.2 

.6 

4.2 

3.6 

% 

21.7 

21.0 

20.3 

■7 

16.8 

16. 1 

15-4 

•7 

4.9 

4-2 

24.8 

24.0 

23.2 

.8 

19.2 

18.4 

17.6 

.8 

5-6 

4.8 

27.9     1      27.0     1    26.1      1      .9  1 

21.6           20.7      1     ig.8      1       .9  1       6.3       1       5.4      1 

87 


29°. 

' 

L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos.     d. 

0 

9.68557 

9.74  375 

30 
30 

0.25  625 

9.94  182 

60 

I 

9 

.68  58o 

23 

9.74405 

0.25  595 

9 

94  175 

59 

2 

9 

.68603 

9.74435 

0.25  565 

9 

94  168 

58 

3 

9 

.68625 

23 

9.74465 

29 

0. 25  535 

9 

94  161 

57 

4 

9 

.68  648 

23 

9.74494 

30 

0.25  5o6 

9 

94  i54 

56 

b 

9 

.68671 

9.74  524 

0.25  476 

9 

94  147 

55 

6 

9 

.68694 

23 

22 

9.74554 

29 

0.25  446 

9 

94  i4o 

54 

7 

9 

.68  716 

9.74  583 

30 

0.25  417 

9 

94  1 33 

53 

8 

9 

.68  739 

9.74  6i3 

0.25  387 

9 

94  126 

52 

9 

9 

.68  762 

23 

9.74  643 

0.25  357 

9 

94  119 

5i 

10 

9 

.68  784 

9.74  673 

0.25  327 

9 

94  1 12 

50 

11 

9 

.68  807 

22 

9.74  702 

30 

0.25  298 

9 

94  105 

49 

12 

9 

.68  829 

23 

9.74  7^2 

30 

0.25  268 

9 

94  098 

48 

i3 

9 

.68  852 

9.74  762 

0. 25  238 

9 

94  090 

47 

23 

29 

i4 

9 

.68875 

22 

9.74791 

30 

0.25  209 

9 

94o83 

46 

i5 

9 

.68  897 

9.74  821 

30 

0.25  179 

9 

94  076 

45 

i6 

9 

.68  920 

9.74  851 

0.25  149 

9 

94  069 

44 

22 

39 

I? 

9 

.68  942 

23 

9.74  880 

30 

0.25   120 

9 

94  062 

43 

i8 

9 

.68  965 

9.74  910 

0.25  090 

9 

94o55 

42 

'9 

9 

.68  987 

9.74939 

30 

0.25  061 

9 

94  o48 

4i 

20 

9 

.69  010 

9.74969 

0.25  o3i 

9 

94  o4i 

40 

21 

9 

.69  o32 

9.74998 

30 

0.25  002 

9 

94  o34 

39 

22 

9 

.69  055 

9.75  028 

0.24  972 

9 

94  027 

38 

23 

9 

.69077 

23 

9.75  o58 

29 

0.24  942 

9 

94  020 

8 

37 

24 

9 

.69  100 

9.75  087 

0.24  913 

9 

94  012 

36 

2b 

9 

.69   122 

9.75  117 

0.24  883 

9 

94  oo5 

35 

26 

9 

.69  i44 

9.75  i46 

29 

0.24  854 

9 

93  998 

y 

34 

27 

9 

69  167 

23 

9.75  176 

0.24  824 

9 

93991 

7 

33 

28 

9 

69  189 

9.75  2o5 

0.24795 

9 

93  984 

32 

29 

9 

69  212 

23 

9.75235 

0.24  765 

9 

93977 

7 

3i 

29 

7 

30 

9 

69  234 

9.75  264 

0.24  736 

9 

93970 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

60°  30 

r. 

1 

PP 

.1 

30 

39 

.1 

~ 

33 

33 

.1 

8 

7 

30 

li 

2-3 

2.3 

0.8 

0-7 

.2 

6.0 

.3 

4.6 

4-4 

.2 

1.6 

1-4 

•3 

9.0 

8.7 

•3 

6.9 

6.6 

•3 

2.4 

2.1 

•4 

12.0 

11.6 

•4 

9.2 

8.8 

•4 

3-2 

3.8 

•5 

15.0 

14.5 

•5 

"•5 

II. 0 

•s 

4.0 

35 

.6 

18.0 

•7-4 

.6 

138 

13.2 

.6 

4-8 

4-2 

•7 

ai.o 

20.3 

.7 

16. 1 

15.4 

.7 

56 

4-9 

.8 

24.0 

23.2 

.8 

.8.4 

.7.6 

.8 

6.4 

5.6 

26.1 

19.8 

^^ 

-^ 

88 


29°  30 . 


' 

L.  Sin. 

d. 

L.  Tang.  ,  d. 

L.  Cotg. 

L.  Cos.     d. 

30 

9.69  234 

22 

9. 75  264 

30 

0.24  736 

9.93970 

7 

R 

30 

29 

3i 

9 

69  256 

9.75  294 

0.24  706 

9.93  963 

32 

9 

69279 

9.75  323 

0.24  677 

9.93  955 

28 

33 

9 

69  3oi 

22 

9.75  353 

30 
29 

0.24  647 

9.93  948 

27 

34 

9 

69  323 

9.75  382 

0.24  618 

9.93  94i 

26 

35 

9 

69345 

9.75  4ii 

0.24  589 

9.93  934 

25 

36 

9 

69368 

9.75  44i 

30 

0.24  559 

9.93927 

24 

3? 

9 

69  390 

9.75  470 

30 

0.24  53o 

9.93  920 

23 

38 

9 

69  4l2 

9.75  500 

0.24  5oo 

9.93  912 

22 

39 

9 

69434 

9.75  529 

29 
29 

0.24  471 

9.93  905 

21 

20 

40 

9 

69456 

9.75  558 

0.24  442 

9.93  898 

4i 

9 

69479 

22 

9.75  588 

29 

0.24  4l2 

9.93  891 

'9 

42 

9 

69  5oi 

9.75  617 

0.24  383 

9.93884 

18 

43 

9 

69523 

9.75  647 

0.24353 

9.93  876 

17 

44 

9 

69545 

9.75  676 

0.24  324 

9.93869 

16 

45 

9 

69  567 

9,75  705 

0.24  295 

9.93  862 

i5 

46 

9 

69  589 

9.75735 

30 

0.24  265 

9.93855 

i4 

47 

9 

69  611 

9.75  764 

0.24  236 

9-93  847 

i3 

48 

9 

69633 

9.75  793 

0.24  207 

9.93  84o 

12 

49 

9 

69655 

9. 75  822 

29 

0.24  178 

9.93833 

II 

50 

9 

69  677 

22 

22 

9.75  852 

29 
29 

0.24  i48 

9 ,93  826 

10 

9 

5i 

9 

69  699 

9.75  881 

0.24  119 

9.93  819 

b2 

9 

69  721 

9.75  910 

0,24  090 

9.93  811 

8 

53 

9 

69743 

9.75939 

0.24  061 

9.93  8o4 

7 

54 

9 

69  765 

9.75969 

29 

0.24  o3i 

9.93797 

6 

55 

9 

69787 

9.75998 

29 

0.24  002 

9.93789 

5 

56 

9 

69  809 

22 

9.76  027 

0.23  973 

9.93  782 

4 

22 

29 

5? 

9 

69831 

9.76  o56 

30 

0.23  944 

9-93775 

3 

58 

9 

69853 

9.76  086 

0.23  914 

9.93  768 

2 

59 

9 

69875 

9.76  115 

0.23  885 

9.93  760 

I 

60 

9 

.69  897 

9.76  i44 

0. 23  856 

9.93  753 

0 

L.  Cos. 

'  d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin.     d. 

' 

60°. 

PP 

.1 

30 

29 

23 

22 

.1 

8 

7 

30 

2.9 

.1 

2.3 

2.2 

0.8 

0.7 

■  2 

6.0 

5-8 

.2 

4.6 

4-4 

.2 

1.6 

1-4 

■3 

9.0 

8.7 

•3 

6.9 

6.6 

•3 

2.4 

2.1 

•4 

12.0 

II. 6 

•4 

9.2 

8  8 

•4 

3-2 

2.8 

•5 

150 

14-5 

•5 

"•5 

II. 0 

■5 

4.0 

3-5 

.6 

18.0 

17-4 

.6 

•3.8 

13-2 

.6 

4.8 

4.2 

•7 

21.0 

20.3 

•7 

16. 1 

iS-4 

.7 

5.6 

4.9 

.8 

24.0 

23.2 

.8 

.8.4 

.7.6 

.8 

6.4 

5-6 

27.0 

26.1 

19.8 

i„^. 

■9            7-2       1        6.3      1 

30°. 


' 

L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotsr. 

L.  Cos.    1  d. 

^^"' 

0 

9.69897 

9.76  i44 

0.23  856 

9.93  753 

60 

I 

9.69919 

9.76  173 

29 

0.23827 

9.93  746 

8 

59 

2 

9.69  941 

9.76  202 

0.23  798 

9.93  738 

58 

3 

9.69  963 

21 

9.76  23l 

3° 

0.23  769 

9.93  73i 

7 

7 

S7 

4 

9.69  984 

9.76  261 

29 

0.23  739 

9.93  724 

56 

b 

9.70  006 

9.76  290 

0.23  710 

9.93717 

8 

bb 

6 

9.70  028 

9.76  319 

29 

0.23  681 

9.93709 

54 

22 

29 

7 

7 

9.70  050 

9.76  348 

29 

0.23  652 

9.93  702 

b3 

8 

9.70  072 

9.76377 

0.23  623 

9.93  695 

8 

32 

9 

9.70  093 

9.76  4o6 

0.23  594 

9.93687 

5i 

10 

9.70  1 15 

9.76435 

0.23  565 

9.93  680 

7 

50 

1 1 

9.70  137 

9.76  464 

29 

0.23  536 

9.93  673 

R 

49 

12 

9.70  169 

9.76  493 

0.23  507 

9.93665 

48 

i3 

9.70  180 

22 

9.76  522 

29 
29 

0.23478 

9.93  658 

7 
8 

47 

i4 

9.70  202 

9.76551 

29 

0.23  449 

9.93  65o 

7 

46 

i5 

9.70  224 

9.76  58o 

0.23  420 

9.93643 

45 

i6 

9.70  245 

21 
22 

9.76  609 

29 
30 

0.23  391 

9.93636 

7 
8 

44 

n 

9.70  267 

9.76  639 

0.23  36I 

9.93  628 

7 

43 

i8 

9.70  288 

9.76  668 

0.23  332 

9.93  621 

42 

'9 

9.70  3io 

22 
22 

9.76697 

29 
28 

0.23  3o3 

9.93  6i4 

7 
8 

4i 
40 

20 

9.70  332 

9.76  725 

0.23  275 

9.93  606 

21 

9.70  353 

9.76  754 

29 

0.23  246 

9.93599 

39 

22 

9.70375 

9.76  783 

0.23  217 

9.93  591 

38 

23 

9.70  396 

22 

9.76  812 

29 

0.23  188 

9.93  584 

7 
7 

37 

24 

9.70  4' 8 

9.76841 

29 

0.23  i59 

9.93^7 

8 

36 

2b 

9.70439 

9.76  870 

0.23  i3o 

9.93  569 

35 

26 

9.70  46i 

21 

9.76899 

29 
29 

0.23  lOI 

9.93  562 

7 
8 

34 

27 

9.70  482 

9.76  928 

0.23  072 

9.93  554 

33 

28 

9.70  5o4 

9.76957 

0. 23  043 

9.93  547 

32 

29 

9.70  625 

9.76  986 

0.23  oi4 

9.93  539 

3i 

29 

7 

30 

9.70  547 

9.77015 

0.22  985 

9.93  532 

30 

L.  Cos. 

d. 

L.Cotg.  !   d. 

L.  Tang. 

L.  Sin. 

d. 

■ 

59°  30 

1 

PP 

30 

»9 

38 

.1 

32 

31 

.1 

8 

7 

30 

2.9 
5.8 

2.8 

2 

2 

2.1 

08 

0.7 

.2 

6.0 

V6 

.2 

4 

4 

4-2 

.2 

1.6 

1.4 

•3 

9.0 

8.7 

8.4 

•3 

6 

6 

6.3 

•3 

24 

2.1 

•4 

I2.0 

I 

1.6 

II. 2 

•4 

8 

8 

8.4 

•4 

3-2 

2.8 

■5 

ISO 

I 

45 

14.0 

•5 

II 

0 

IO-5 

•5 

4.0 

3-5 

.6 

18.0 

I 

74 

16.8 

.6 

13 

.2 

12.6 

.6 

4.8 

4.2 

•7 

21.0 

2 

0-3 

19.6 

•7 

i.S 

4 

14.7 

.7 

5-6 

4.9 

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24.0 

2 

3.2 

22.4 

.8 

17 

6 

16.8 

.8 

6.4 

5-6 

•q    1      27.0     1      2 

6  I          25.2    ! 

-^ 

8 

18.9 

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90 


30 

°30 

• 

/ 

L.  Sin. 

d.     ] 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.70  547 

9.77015 

29 
29 

0.22  985 

9.93  532 

30 

31 

9 

70  568 

22 

9.77  o44 

0.22  956 

9.93  525 

8 

29 

32 

9 

70  590 

9.77073 

28 

0.22  927 

9.93517 

28 

33 

9- 

70  61 1 

22 

9.77  101 

29 

0.22  899 

9.93  5io 

8 

27 

34 

9 

70  633 

21 

9.77  i3o 

29 

0.22  870 

9.93  502 

26 

35 

9 

70  654 

9.77  159 

29 

0.22  84i 

9.93495 

g 

25 

36 

9- 

70  675 

y.77  188 

0.22  812 

9.93  487 

24 

22 

29 

7 

3? 

9- 

70  697 

21 

9.77217 

29 

0.22  783 

9.93480 

8 

23 

38 

9 

70  718 

9.77  246 

0.22  754 

9.93  472 

22 

39 

9 

70  739 

9.77274 

0.22  726 

9.93  465 

8 

21 

40 

9 

70  761 

9.77  3o3 

29 

29 
29 

0.22  697 

9.93  457 

7 
8 

20 

19 

4i 

9 

70  782 

21 

9.77  332 

0.22  668 

9.93450 

42 

9 

70  8o3 

9.77  36i 

0.22  639 

9.93  442 

18 

43 

9 

70  824 

9.77390 

28 

0.22  610 

9.93435 

R 

17 

44 

9 

70  846 

21 

9.77418 

29 

0.22  582 

9.93  427 

7 

16 

45 

9 

70  867 

9.77447 

0.22  553 

9.93  420 

3 

i5 

46 

9 

70888 

9.77476 

29 

0.22  524 

9.93  4l2 

i4 

47 

9 

70  909 

22 

9.77505 

28 

0.22  495 

9.93  405 

8 

i3 

48 

9 

70  931 

9.77  533 

0.22  467 

9.93  397 

12 

49 

9 

70  952 

9.77  562 

29 
29 
28 

0.22 438 

9.93   390 

8 

II 

50 

9 

70973 

9.77591 

0.22  409 

9.93  382 

10 

5i 

9 

70  994 

9.77619 

29 

0.22  38i 

9.93  375 

8 

9 

52 

9 

71  oi5 

9.77  648 

0.22  352 

9.93  367 

7 
8 

8 

53 

9 

71  o36 

9.77677 

29 

0.22  323 

9.93  36o 

7 

22 

29 

54 

9 

71  o58 

9.77706 

28 

0.22  294 

9.93  352 

8 

6 

55 

9 

71  079 

9.77734 

0.22  266 

9.93344 

7 
8 

5 

56 

9 

71  100 

9.77763 

29 
28 

0.22  237 

9.93  337 

4 

57 

9 

71  121 

9.77791 

29 

0.22  209 

9.93  329 

7 

3 

58 

9 

71   l42 

9.77  820 

0.22  180 

9.93  322 

8 

2 

59 

9 

71  i63 

9.77849 

29 

28 

0.22  i5i 

9.93  3i4 

7 

I 

60 

9 

.71  i84 

9.77877 

0.22    123 

9.93  307 

0 

L.  Cos. 

d. 

L.  Cotg.  1  d. 

L.  Tang. 

L.  Sin.    id. 

/ 

59°. 

1 

PP 

.1 

29 

28 

.1 

32 

31 

.1 

8 

7 

2.9 

2.8 

2.2 

2.1 

0.8 

0.7 

.2 

5.8 

5-6 

.2 

4.4 

4.2 

.2 

1.6 

1.4 

•3 

8.7 

8.4 

•3 

6.6 

6-3 

•3 

24 

2.1 

•4 

11.6 

II. 2 

4 

8.8 

8.4 

•4 

3-2 

28 

•S 

14.5 

14.0 

•s 

II. 0 

1C.5 

■  5 

4.0 

3  5 

.6 

17.4 

16.8 

.6 

13.2 

12.6 

.6 

4.8 

4.2 

.7 

20.3 

19.6 

•7 

iS-4 

14.7 

•7 

5-6 

4.9 
5-6 

6.3 

.8 

23.2 

22.4 

.8 

17.6 

16.8 

.8 

6.4 

1             9 

26.1 

25.2 

18.9 

^^ 

— 1. 

91 


31°. 


' 

L.Sin.    1  d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

^■^ 

0 

9.71  184 

9.77877 

0.22   123 

9.93  807 

60 

I 

9.71  205 

21 

9.77906 

29 

0.22  094 

9.93299 

8 

59 

2 

9.71  226 

9-77  9'''5 

28 

0.22  o65 

9.93  291 

58 

3 

9.71  247 

21 

9.77963 

29 

0.22  037 

9.93284 

1 
8 

57 

4 

9.71  268 

21 

9.77992 

28 

0.22  008 

9.93  276 

7 

56 

5 

9.71  289 

9.78  020 

0.21  980 

9.93  269 

55 

6 

9.71  3io 

9.78  049 

28 

0.21  951 

9.93  261 

54 

7 

9.71  33i 

21 

9.78077 

29 

0.21  923 

9.93  253 

53 

8 

9.71  352 

9.78  106 

0.21  894 

9.93  246 

52 

9 

9.71  373  j    "   1 

9.78135 

28 
29 
28 

0.21  865 

9.93238 

8 

7 
3 

5i 

10 

9.71  393 

2r 
21 

9.78  i63 

0.21  837 

9.93  23o 

50 

1 1 

9.71  4i4 

9.78  192 

0.21  808 

9.93  223 

49 

12 

9.71  435 

9.78  220 

29 
28 

0.21  780 

9 .  93  2 1 5 

48 

i3 

9.71  456 

21 

9.78249 

0.21  75i 

9.93  207 

8 

47 

i4 

9.71  477 

21 

9.78277 

29 

0.21  723 

9.93  200 

46 

i5 

9-71  49^ 

9.78  3o6 

28 

0.21  694 

9.93  192 

45 

i6 

9.71  519 

20 

9.78  334 

29 

0.21  666 

9.93  184 

44 

17 

9.71  539 

21 

9.78363 

28 

0.21  637 

9.93  177 

8 

43 

i8 

9.71  56o 

9.78  391 

28 

0.21  609 

9.93  169 

42 

'9 

9.71  5Si 

21 

9.78  419 

29 

28 

0.21  58i 

9.93  161 

7 

4i 

20 

g.71  602 

9.78448 

0.21  552 

9.93  i54 

40 

21 

9.71  622 

9.78476 

29 

0.21  524 

9.93  i46 

8 
8 

39 

22 

9.71  643 

9.78505 

28 

0.21  495 

9.93  i38 

38 

2  3 

9.71  664 

21 

9.78533 

29 

0.21  467 

9.93  i3i 

7 
8 

37 

24 

9.71685 

9.78  562 

28 

0.21  438 

9.93   123 

8 

36 

2D 

9.71  705 

9.78  590 

28 

0.21  4io 

9.93  ii5 

35 

26 

9.71  726 

21 

9.78618 

29 

0.21  382 

9.93  108 

7 
8 

34 

27 

9.71  747 

9.78647 

28 

0.21  353 

9.93  100 

g 

33 

28 

9.71  767 

9.78675 

0.21  325 

9.93  092 

8 
7 

32 

29 

9.71  78S 

21 

9.78  704 

28 

0.21  296 

9.93084 

3i 
30 

30 

9.71  809 

9.78  732 

0.21  268 

9.93077 

L.Cos.    i   d.  1 

L.  Cotg.      d. 

L.  Tang. 

L.Sin. 

d. 

' 

58°  30 .                                                    1 

PP 

39 

38 

21 

20 

8 

7 

.1 

2.9 

2.8 

.1 

2.1 

2.0 

.1 

0.8 

0.7 

.2 

5-8 

5-6 

.2 

4-2 

4.0 

.2 

1.6 

1.4 

•  3 

8.7 

8.4 

•3 

6.3 

6.0 

■3 

24 

2.1 

•  4 

11.6 

II. 2 

•4 

8.4 

8.0 

.4 

3.2 

2.8 

■5 

MS 

14.0 

•s 

10.5 

10.0 

•  ,s 

4.0 

3-5 

.6 

17-4 

16.8 

.6 

12.6 

12.0 

.6 

4-8 

4.2 

•7 

20.3 

19.6 

•7 

14.7 

140 

■7 

5-6 

49 

.8 

23.2 

22.4 

.8 

16.8 

16.0 

.8 

6.4 

5.6 

26. 1 

25.2 

18.9 

18.0 

_6^ 

"i?- 


31 

°30 

', 

/ 

L.  Sin.    i  d. 

L.  Tang-. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.71  809 

9.78  732 

28 
29 
28 

0.21  268 

9.98077 

8 

30 

3i 

9 

.71  829 

9.78  760 

0.21  240 

9.93  069 

8 

29 

32 

9 

.71  850 

9.78  789 

0.21    211 

9.98  061 

8 

28 

33 

9 

.71  870 

9.78  817 

0.21  i83 

9.98053 

27 

21 

28 

7 

34 

9 

.71  891 

9.78845 

29 
28 
28 

0.21  155 

9.98  o46 

8 

26 

3  b 

9 

.71  911 

9.78874 

0.21  126 

9.98088 

8 

25 

36 

9 

.71  982 

20 

9.78  902 

0.21  098 

9.98  080 

8 

24 

:^7 

9 

.71  952 

9.78  980 

29 
28 
28 

28 
29 
28 

28 

0.21  070 

9.98  022 

8 

23 

38 

9 

.71  973 

9.78  959 

0.21  o4i 

9.98  oi4 

7 

22 

39 

9 

.71  994 

9.78987 

0.21  oi3 

9.98  007 

21 

40 

9 

.72  oi4 

9.79  oi5 

0.20  985 

9.92999 

8 

20 

4i 

9 

.72  o34 

9.79  043 

0.20  957 

9.92991 

8 

'9 

42 

9 

.72  055 

9.79072 

0.20  928 

9.92  988 

7 
8 

18 

43 

9 

.72  075 

21 

9.79  100 

0.20  900 

9.92976 

17 

44 

9 

.72  096 

9.79  128 

28 

0.20  872 

9.92  968 

8 

16 

4b 

9 

.72  116 

9.79  i56 

0.20  844 

9.92  960 

8 

i5 

46 

9 

.72  137 

20 

9.79  185 

29 
28 

0.20  81 5 

9.92  952 

8 

i4 

47 

9 

.72  i57 

20 

9.79  2l3 

28 

0.20  787 

9.92  944 

8 

i3 

4« 

9 

.72  177 

9.79  241 

28 
28 

0.20  759 

9.92  986 

7 

12 

49 

9 

.72  198 

9.79269 

0.20  781 

9.92929 

1 1 

50 

9 

.72  218 

9.79297 

0.20  708 

9.92  921 

8 

10 

29 

5i 

9 

72  238 

9.79  326 

28 

0.20  674 

9.92  913 

8 

9 

52 

9 

72  259 

9.79  354 

28 
28 

0.20  646 

9.92  905 

8 

8 

53 

9 

72  279 

9.79  382 

0.20  618 

9.92897 

8 

7 

54 

9 

72  299 

9.79  4io 

28 

0.20  590 

9.92  889 

8 

6 

55 

9 

72   320 

9.79438 

0.20  562 

9.92881 

5 

56 

9 

72  34o 

20 
20 

9.79  466 

29 

,0.20  534 

9.92  874 

8 

4 

^7 

9 

72  36o 

9.79495 

28 

0.20  5o5 

9.92  866 

8 

3 

58 

9 

72  38i 

9.79  523 

28 
28 

0.20  477 

9.92  858 

8 

2 

59 

9 

.72  4oi 

9.79  55i 

0.20  449 

9.92  850 

8 

I 

60 

9 

72  421 

9.79  579 

0.20  421 

9.92  842 

0 

L.  Cos.    1  d.  1 

L.  Cotg.  !   d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

58°.                                                         1 

PP 

.  I 

29 

28 

.1 

21 

20 

.1 

8 

7 

2.9 

2.8 

2.1 

2.0 

0.8 

0.7 

.2 

S-8 

5-6 

.2 

4.2 

4.0 

.2 

1.6 

1.4 

•3 

8.7 

8.4 

•3 

6-3 

6.0 

■3 

2.4 

2.1 

•4 

II. 6 

II. 2 

•4 

8.4 

8.0 

•4 

3-2 

2.8 

•5 

145 

14.0 

•5 

10.5 

10. 0 

•  s 

4.0 

3-5 

.6 

17.4 

16.8 

.6 

12.6 

12.0 

.6 

4.8 

4.2 

.7 

20.3 

19.6 

•7 

14.7 

14.0 

.7 

S-6 

4.9 

.8 

23.2 

22.4 

.8 

16.8 

16.0 

.8 

6.4 

5-6 

26.1 

18.9 

18.0 

^^ 

-:2- 

6-3 

93 


32°. 


~o" 

L.  Sin. 

d. 

L.  Tang.     d.  ]  L.  Cotg.  | 

L.  Cos.     d.| 

9.72  421 

9.79579 

28 
28 

0.20  421 

9.92  842 

60 

I 

9.72  44i 

20 

9.79607 

0.20  393 

9.92834 

8 

59 

2 

9.72  461 

9.79  635 

28 
28 

0.20  365 

9.92  826 

8 

58 

■i 

9.72  482 

20 

9.79  663 

0.20  337 

9.92  818 

8 

i>7 

4 

9.72  5o2 

20 

9.79691 

28 

0.20  809 

9.92  810 

7 

56 

5 

9.72   522 

9.79719 

28 

0.20  281 

9.92  8o3 

8 

bb 

6 

9.72  542 

20 

9.79747 

29 

0.20  253 

9.92795 

8 

b4 

7 

9.72  562 

20 

9.79776 

28 

0.20  224 

9.92787 

8 

53 

8 

9.72  582 

9.79  8o4 

28 

0.20  196 

9.92779 

R 

52 

9 

9.72  602 

9.79  832 

28 
28 

28 

0.20  168 

9.92  771 

8 

5i 

10 

9.72  622 

9.79  860 

0.20  i4o 

9.92  763 

8 
8 

50 

1 1 

9.72  643 

20 

9.79888 

0.20  112 

9.92  755 

49 

12 

9.72  663 

9.79916 

28 

0.20  o84 

9.92  747 

8 

48 

i3 

9.72683 

9.79944 

0.20  o56 

9.92  739 

47 

20 

28 

8 

i4 

9.72  703 

20 

9.79972 

28 

0.20  028 

9.92  781 

8 

4b 

lb 

9.72  723 

9.80  000 

0.20  000 

9.92  723 

8 

4b 

i6 

9.72  743 

9.80  028 

28 

0.19  972 

9.92  7i5 

44 

20 

28 

8 

17 

9.72  763 

9.80  o56 

28 

0.19  944 

9.92707 

8 

43 

i8 

9.72  783 

9.80084 

0.19  916 

9.92699 

8 
8 

42 

•9 

9.72  8o3 

9.80  112 

0.19888 

9.92  691 

4i 

20 

9.72  823 

9.80  i4o 

28 

28 

0. 19  860 

9.92  683 

40 

21 

9,72  843 

9.80  168 

27 
28 

0.19  832 

9.92  675 

8 

39 

22 

9.72  863 

9.80  195 

0. 19  805 

9.92  667 

8 

38 

23 

9.72  883 

9.80223 

0.19777 

9.92  659 

37 

'9 

28 

8 

24 

9.72  902 

9.80  25l 

28 

0.19  749 

9.92  65i 

8 

36 

2b 

9.72  922 

9.80  279 

28 

0. 19  721 

9.92  643 

8 

3b 

26 

9.72  942 

9 .  80  3(^7 

0. 19  693 

9.92  635 

34 

20 

28 

8 

27 

9.72  962 

9.80335 

28 

0.19665 

9.92  627 

8 

33 

28 

9.72  982 

9.80  363 

28 
28 

0.19  637 

9.92  619 

8 
8 

32 

29 

9.73  002 

9.80  391 

0.19  609 

9.92  61 1 

3i 

30 

9.73  022 

9.80  419 

0.19  58i 

9.92  6o3 

30 

L.  Cos. 

d. 

L.  Cotg.  1   d. 

L.  Tang. 

L.  Sin. 

d? 

' 

67°  30 .                                                1 

PP 

ag 

38 

27 

.1 

ai 

30 

19 

8 

7 

2.9 

2.8 

27 

2.1 

2.0 

1. 9 
38 

.1 

08 

0.7 

.2 

§•** 

5-6 

I* 

.3 

4.2          4.0 

.3 

1.6 

>  4 

■3 

8.7 

8.4 

8.1 

3 

6-3 

6.0 

5-7 

•3 

2.4 

2.1 

•4 

H.6 

II. 2 

108 

•4 

8.4 

8.0 

7.6 

•4 

3-3 

2.8 

•5 

M-5 

14.0 

>3-5 

•5 

10.5 

lO.O 

9  5 

•s 

4.0 

3-5 

.6 

'7-4 

16.8 

l6.2 

.6 

12.6 

12.0 

11.4 

.6 

4.8 

4-3 

•7 

ao.3 

19.6 

18.9 

•7 

'f-Z 

140 

133 

•7 

5-6 

4-9 

.8 

23.2 

22.4 

21.0 

.8 

16.8 

16.0 

15.2 

.8 

6.4 

5-6 

26.1 

25.2 

•9 

18.9 

18.0 

.— &. 

7' 

^•3 

94 


32 

°30 

• 

' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.73  022 

19 
20 

9.80  419 

28 
27 

0. 19  58i 

9.92  6o3 

8 
8 

30 

3i 

9.73  o4i 

9.80447 

0. 19  553 

9.92595 

29 

32 

9.73  061 

9.80474 

28 

0.19  526 

9.92  587 

28 

33 

9.73081 

20 

9.80  5o2 

28 

0. 19498 

9.92579 

8 

27 

34 

9.73  lOI 

9.80  53o 

28 

0. 19470 

9.92  571 

8 

26 

35 

9.73   121 

9.80  558 

0.19442 

9.92  563 

25 

36 

9.73  i4o 

19 

9.80  586 

28 

0.19414 

9.92555 

M 

3? 

9.73   160 

9.80  6i4 

28 

0.19386 

9.92  546 

R 

23 

38 

9.73   180 

9.80  642 

0.19358 

9.92538 

8 
8 

8 

8 

22 

39 

9.73  200 

•9 
20 

9.80  669 

27 
28 

28 
28 

0.19331 

9.92  53o 

21 

40 

9.73  219 

9.80  697 

0.19303 

9.92  522 

20 

19 

4i 

9.73  239 

9.80  725 

0. 19  275 

9.92  5i4 

42 

9.73  259 

9.80  753 

0. 19  247 

9.92  5o6 

18 

43 

9.73  278 

19 
20 

9.80781 

27 

0. 19  219 

9.92498 

8 

17 

44 

9.73  298 

9.80808 

28 

0.19 192 

9.92  490 

8 

16 

45 

9.73318 

9.80  836 

28 
28 

0.19  i64 

9.92  482 

i5 

46 

9.73337 

19 
20 

9.80864 

0.19 i36 

9.92473 

8 

i4 

47 

9.73  357 

9.80  892 

0.19 108 

9.92  465 

8 

i3 

48 

9.73377 

9.80  919 

28 
28 

28 

0.19 081 

9.92  457 

3 

12 

49 

9.73396 

19 
20 

•9 

9.80947 

0.19053 

9.92  449 

8 

8 
8 

II 
10 

50 

9.73416 

9.80975 

0.  19  025 

9.92  44i 

5i 

9.73435 

9.81  oo3 

0.18997 

9.92433 

9 

52 

9.73455 

9.81  o3o 

0.18  970 

9.92425 

8 

53 

9-73474 

19 
20 

9.81  o58 

28 

28 

0.18942 

9.92  4i6 

y 

8 

7 

54 

9.73494 

9.81  086 

0.18  914 

9.92  4o8 

8 

6 

55 

9.73  5i3 

9.81  ii3 

0.18887 

9.92  4oo 

5 

56 

9.73533 

9.81  i4i 

28 

0.18859 

9.92  392 

3 

4 

5? 

9.73  552 

•9 

9.81  169 

o.i883i 

9.92  384 

R 

3 

58 

9.73  572 

9.81  196 

27 

0.18  8o4 

9.92  376 

2 

59 

9.73591 

19 

9.81  224 

28 

0.  i8  776 

9.92  367 

9 

I 

60 

9.73  61 1 

9.81  252 

0.18  748 

9.92  359 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

57°.                                                   1 

PP 

.1 

38 

27 

20 

19 

.1 

9 

8 

2.8 

2.7 

.1 

2.0 

1.9 

0.9 

.0.8 

.2 

5.6 

5-4 

.2 

4.0 

3- 8 

.2 

1.8 

1.6 

•3 

8.4 

8.1 

•3 

6.0 

5-7 

■3 

2.7 

2.4 

•4 

II  2 

10.8 

•4 

8.0 

7.6 

•4 

3-6 

3.2 

•5 

14.0 

13-5 

•5 

10.0 

95 

•5 

4-5 

4.0 

.6 

16.8 

16.2 

.6 

12.0 

11.4 

.6 

5-4 

4.8 

■7 

19.6 

18.9 

■7 

14.0 

13-3 

•7 

6-3 

5-6 

.8 

22.4 

21.6 

.8 

16.0 

IS-2 

.8 

7.2 

6.4 

18.0 

.9    1       8.1 

95 


33°. 


' 

L.Sin.    1   d. 

L.  Tang.  \  d. 

L.  Cotg. 

L.  Cos.    ;  d. 

0 

9.73  6n 

9.81   252 

27 
28 

0.18  748 

9.92  359 

8 
8 

60 

I 

9.73  63o 

20 

9.81   279 

0.18  721 

9.92  35i 

59 

2 

g.73  650 

9.81    307 

28 

0.18  693 

9.92  343 

58 

3 

9.73  669 

30 

9.81   335 

27 

O.I8665 

9.92335 

q 

57 

4 

9.73  689 

9.81  362 

38 

0.18  638 

9.92  326 

8 

56 

5 

9.73  708 

9.81  390 

38 

0.18  610 

9.92  3i8 

8 

55 

6 

9.73  727 

9.81  4i8 

0.18 582 

9.92  3io 

54 

20 

27 

8 

7 

9.73  747 

9.81445 

28 

0.18  555 

9.92  3o2 

53 

8 

9.73  766 

9  81473 

0.18  527 

9.92  293 

52 

9 

9.73785 

'9 

9.81  5oo 

28 

0. 18  500 

9.92  285 

5i 

10 

9.73  805 

'9 
'9 

9.81  528 

28 
27 

0. 18  472 

9.92277 

8 

9 
8 

50 

1 1 

9.73  824 

9.81  556 

0.18444 

9.92  269 

49 

12 

9.73843 

9.81  583 

0. 18  417 

9.92  260 

48 

i3 

9.73863 

9.81  611 

0.18  389 

9.92  252 

47 

'9 

27 

8 

i4 

9.73882 

9.81  638 

28 

0.18  362 

9.92  244 

9 
8 
8 

46 

i5 

9.73  901 

9.81  666 

0.18334 

9.92  235 

45 

i6 

9.73  921 

19 

9.81  693 

28 

0.18  307 

9.92  227 

44 

I? 

9.73  940 

19 

9.81  721 

0. 18  279 

9.92  219 

8 

43 

i8 

9.73959 

9.81  748 

0.  l8  252 

9.92  211 

42 

•9 

9.73978 

9.81  776 

0. 18  224 

9.92  202 

y 

4i 

20 

9.73  997 

9.81  8o3 

27 
28 

0. 18  197 

9.92  194 

3 

40 

21 

9.74017 

9.81  83i 

0.18  169 

9.92  186 

9 

39 

22 

9.74  o36 

9.81  858 

28 

0.18  l42 

9.92  177 

38 

23 

9.74  o55 

'9 

9.81  886 

0.18  ii4 

9.92  169 

37 

19 

27 

24 

9.74  074 

'9 

9.81  913 

28 

0.18  087 

9.92  161 

36 

2b 

9.74  093 

9.81  941 

0. 18  059 

9.92  l52 

35 

26 

9.74  ii3 

19 
19 

9.81  968 

27 

o.i8o32 

9.92  144 

ft 

34 

27 

9.74  l32 

9.81  996 

0.18  oo4 

9.92  i36 

g 

33 

28 

9.74  i5i 

9.82  023 

28 

0.17977 

9.92  127 

g 

32 

29 

9.74  170 

9.82  o5i 

0.17  949 

9.92  1 19 

3 

3i 

30 

9.74  189 

9.82  078 

27 

0. 17  922 

9.92  III 

30 

L.  Cos. 

d." 

L.  Cotg. 

d. 

L.Tang.      L.Sin.    | 

d. 

' 

66°  30 

1 

pp 

.1 

.2 

38 

37 

30 

19 

.1 
.3 

9 

8 

2.8 
5.6 

2.7 

5-4 

.1 
.3 

2.0 
4.0 

It 

::t 

0.8 
1.6 

•3 

8.4 

8.1 

.3 

6.0 

5.7 

•3 

2.7 

2-4 

•4 

II. 2 

10.8 

•4 

8.0 

7.6 

•4 

3.6 

3-3 

•5 

14.0 

13s 

•5 

lO.O 

9-5 

.5 

4-5 

4.0 

.6 

16.8 

l6.2 

.6 

12.0 

11.4 

.6 

5-4 

4.8 

•7 

19.6 

18.9 

•7 

14.0 

13.3 

•7 

6.3 

5.6 

.8 

22.4 

21.6 

.8 

16.0 

»5-2 

.8 

7.2 

6.4 

.g            18.0 

.g           8.1 

7.2 

96 


33°  30'. 


L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

"^^ 

30 

9.74  189 

'9 
'9 

9.82  078 

28 
27 

0.17  922 

9.92  III 

9 

30 

29 

3i 

9 

.74  208 

9.82  106 

0.17  894 

9.92  102 

32 

9 

.74  227 

9.82  i33 

28 

0.17  867 

9.92094 

8 

28 

33 

9 

.74  246 

9.82  161 

0.17  839 

9.92  086 

27 

19 

27 

9 

34 

9 

.74  265 

19 

9.82  188 

27 

0.17  812 

9.92077 

g 

26 

35 

9 

.74284 

9.82  2l5 

28 

0.17785 

9.92  069 

25 

36 

9 

.74303 

'9 

9.82243 

27 

0.17757 

9.92  060 

9 
8 

24 

37 

9 

.74322 

19 

9.82  270 

28 

0.17  73o 

9.92  o52 

R 

23 

38 

9 

.74341 

9.82  298 

0.17  702 

9.92  o44 

22 

39 

9 

.74  36o 

'9 

9.82  325 

0.17675 

9.92  o35 

y 

8 

9 

21 

40 

9 

.74379 

19 

'9 
'9 

9.82352 

28 
27 

0.17  648 

9.92  027 

20 

19 

4i 

9 

.74398 

9.82  38o 

0. 17  620 

9.92  018 

42 

9 

74417 

9.82  407 

28 

0.17  593 

9.92  010 

18 

43 

9 

.74  436 

'9 

9.82435 

0.17  565 

9.92  002 

'7 

44 

9 

74455 

19 

9.82  462 

0.17538 

9.91  993 

8 

16 

4b 

9 

74474 

9.82489 

0. 17  5ii 

9.91  985 

i5 

46 

9 

74493 

»9 
19 

9.82  5i7 

27 

0. 17483 

9.91  976 

y 

8 

i4 

47 

9 

74  5 12 

19 
18 

'9 

9.82  544 

27 

28 
27 

0.17  456 

9.91  968 

9 
8 

9 

i3 

48 

9 

74531 

9.82  571 

0. 17  429 

9.91  959 

12 

49 

9 

74  549 

9.82  599 

0. 17  4oi 

9.91  951 

1 1 

50 

9 

74  568 

9.82  626 

0.17  374 

9.91  942 

10 

5i 

9 

74587 

'9 

9.82  653 

27 
28 

0.17  347 

9.91  934 

9 

52 

9 

74  606 

'9 

9.82681 

0.17  319 

9.91  925 

8 

53 

9 

74625 

19 

9.82  708 

27 

0. 17  292 

9.91  917 

7 

54 

9 

74  644 

»9 

9.82  735 

27 
27 
28 

0. 17  265 

9.91  908 

9 

8 

6 

55 

9 

74  662 

9.82  762 

0.17238 

9.91  900 

5 

56 

9 

74681 

19 

9.82  790 

0.17  210 

9.91  891 

y 

4 

57 

9 

74  700 

19 

9.82817 

27 

0.17  i83 

9.91  883 

9 

8 
9 

3 

58 

9 

74719 

19 

9.82844 

0.17  i56 

9.91  874 

2 

59 

9 

74737 

18 
»9 

9.82  871 

27 
28 

0. 17  129 

9.91  866 

I 
0 

60 

9 

74  756 

9.82  899 

0. 17  lOI 

9.91  857 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

56°. 

1 

PP 

.1 

38 

27 

19 

18 

.1 

9 

8 

2.8 

2-7 

.1 

It 

1.8 

n 

0.8 

.2 

5-6 

5-4 

.2 

36 

.2 

1.6 

•3 

8.4 

8.1 

•3 

5-7 

5-4 

•3 

2.7 

2.4 

•4 

II. 2 

ia8 

•4 

7.6 

7.2 

•4 

3-6 

3-2 

•5 

14.0 

«3-5 

•5 

9-5 

9.0 

•5 

4-5 

4.0 

.6 

16.8 

16.2 

.6 

11.4 

10.8 

.6 

5-4 

4.8 

•  7 

19.6 

18.9 

.7 

•3-3 

12.6 

•7 

6.3 

5-6 

.8 

22.4 

21.6 

.8 

15-2 

14.4 

.8 

7.2 

6.4 

25,2    .    _ 

16.2 

.9           8.1 

97 


34°. 


, 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos.     d. 

0 

9.74  766 

19 
•9 

9.82  899 

27 
37 

0.17  lOI 

9.9 

I  857 

8 

60 

I 

9 

•74775 

9.82  926 

0.17  074 

9 

9 

I  849 

59 

2 

9 

.74  794 

18 

9.82953 

38 

0. 17  047 

9 

9 

I  84o 

8 

58 

3 

9 

.74812 

»9 

9.82  980 

0. 17  020 

9 

9 

I  832 

9 

^^7 

4 

9 

.74  831 

19 

9.83  008 

37 

0.16  992 

9 

9 

I  823 

a 

56 

6 

9 

74850 

18 

9.83  035 

0. 16  965 

9 

9 

18.5 

55 

6 

9 

.74868 

9.83062 

0.16  938 

9 

9 

I  806 

8 

64 

7 

9 

74887 

19 

9.83  089 

28 

0. 16  911 

9 

9 

1798 

9 

53 

8 

9 

74  906 

18 

9.83  117 

O.I6883 

9 

9 

I  789 

62 

9 

9 

.74  924 

9.83  i44 

0.16  856 

9 

9 

I  781 

5i 

10 

9 

74943 

18 
•9 

9.83  171 

27 
27 

0. 16  829 

9 

9 

I  772 

9 
8 

50 

1 1 

9 

74  961 

9.83  198 

0.16  802 

9 

9 

I  763 

49 

12 

9 

74  980 

9.83  225 

o.i6  775 

9 

9 

17S5 

48 

i3 

9 

74999 

18 

9.83  252 

28 

o.i6  748 

9 

9 

I  746 

9 
8 

47 

i4 

9 

75  017 

19 

9.83280 

27 

0. 16  720 

9 

9 

I  738 

46 

i5 

9 

75  o36 

18 

9.83  307 

0.  i6  693 

9 

9 

I  729 

46 

i6 

9 

75  o54 

19 

9.83334 

27 

0.16  666 

9 

9 

I  720 

9 
8 

44 

17 

9 

75  073 

18 

9.83  36i 

27 

0. 16  639 

9 

9 

I  712 

43 

18 

9 

75  091 

9.83  388 

0. 16  612 

9 

9 

I  703 

8 
9 

42 

19 

9 

75  1 10 

18 

9.83  4i5 

27 
28 

0.16  585 

9 

9 

I  695 

4i 

20 

9 

75  128 

9.83442 

0.16558 

9 

9 

I  686 

40 

21 

9 

75  147 

18 

9.83  470 

27 

0. 16  53o 

9 

9 

I  677 

8 

39 

22 

9 

75  i65 

9.83497 

0. 16  5o3 

9 

9 

I  669 

38 

2  3 

9 

75i84 

18 

9.83524 

27 

0.16  476 

9 

9 

I  660 

9 
9 

37 

24 

9 

75  202 

9.83  55i 

0.16  449 

9 

9 

I  65i 

8 

36 

25 

9 

75  221 

9.83578 

0. 16  422 

9 

9 

I  643 

35 

26 

9 

75  239 

»9 

9.83  6o5 

27 
27 

0.16  395 

9 

9 

I  634 

y 

9 

34 

27 

9 

75  258 

18 

9.83  632 

0.16  368 

9 

9 

I  625 

g 

33 

28 

9 

75  276 

9.83  659 

0.16  341 

9 

9 

I  617 

32 

29 

9 

75  294 

19 

9.83  686 

27 
27 

0.16  3i4 

9 

9 

I  608 

9 
9 

3i 
30 

30 

9 

75  3i3 

9.83  713 

0. 16  287 

9 

9 

I  599 

L.  Cos. 

d. 

L.  Cotgr. 

d. 

L.  Tangr. 

L. 

Sin. 

d. 

/ 

55°  3C 

►'. 

1 

PP 

.1 

33 

37 

19 

18 

.1 

9 

8 

2.8 

2.7 

.1 

3I 

1.8 

ti 

0.8 

.2 

S-6 

5-4 

.a 

36 

.2 

1.6 

3 

8.4 

8.1 

•3 

5-7 

5-4 

•3 

3-7 

2.4 

•4 

II. 2 

10.8 

•4 

7.6 

7.2 

•4 

3-6 

3-3 

•5 

14.0 

»3S 

•5 

9-5 

9.0 

•5 

4-5 

4.0 

.6 

16.8 

16.3 

.6 

II. 4 

10.8 

.6 

5-4 

4.8 

•7 

19.6 

18.9 

.7 

1.3-3 

12.6 

•7 

6.3 

5.6 

.8 

22.4 

21.6 

.8 

15.2 

14.4 

.8 

7.2 

6.4 

16.2 

■9 

8.1 

7-2 

98 


34 

0 

30 

', 

/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

1 

..  Cos. 

d. 

30 

9.75  3i3 

.R 

9.83  713 

27 
28 
27 
27 

0.16  287 

9 

91  599 

8 
9 
9 
8 

30 

3i 

3-2 

33 

9.75  33i 
9.75  350 
9.75  368 

19 
18 

18 

9.83  740 
9.83  768 
9.83  795 

0.16  260 

0.16  232 

0.16  2o5 

9 
9 
9 

91  591 
91  582 
91573 

29 
28 
27 

34 
35 
36 

9.75  386 
9.75405 
9.75  423 

'9 
18 

t8 

9.83822 
9.83  849 
9.83876 

27 
27 
27 
27 
27 
27 

27 
27 

27 

0.16   178 

0.16  i5i 
0.16  124 

9 
9 
9 

91  565 
91  556 
91  547 

9 
9 
9 
8 

9 
9 
8 
9 
9 
9 
8 

9 
9 
9 
9 
8 

9 
9 
9 
9 

26 

25 

24 

3? 
38 
39 

9.75  44 1 
9.75  459 
9-7^  478 

18 

>9 
18 

18 

19 
18 
t8 

9.83  903 
9.83  930 
9-83  957 

0. 16  097 
0. 16  070 
0.16  o43 

9 
9 
9 

91  538 
91  53o 

91    521 

23 
22 
21 

40 

9.75  496 

9.83984 

0.16  016 

9 

91  5l2 

20 

4i 

42 

43 

9.75  5i4 
9.75  533 
9.75  55i 

9.84  01 1 
9.84  o38 
9.84  065 

0.  i5  989 
0.  i5  962 
o.i5  935 

9 
9 
9 

91  5o4 

91  495 
91  486 

'9 

18 

17 

44 
45 
46 

9.75  569 
9.75587 
9.75  6o5 

18 
18 

19 
18 
18 
18 

18 
18 

'9 
16 

9.84  092 
9.84  1 19 
9.84  i46 

27 
27 

0.  i5  908 
o.i588i 
O.I5854 

9 
9 
9 

91  477 
91  469 
91  460 

16 
i5 
i4 

47 
48 

49 

9.75  624 
9.75  642 
9.75  660 

9.84  173 
9.84  200 
9.84  227 

27 

27 

27 
27 

26 
27 
27 

0.  i5  827 
0.  i5  800 
0.  i5  773 

9 
9 
9 

9i45i 
91  442 
91  433 

i3 
12 
1 1 

10 

50 

9.75  678 

9.84  254 

0.  i5  746 

9 

91  425 

5i 

52 

53 

9.75  696 
9.75  714 
9.75  733 

9.84280 
9.84  3o7 
9.84334 

0.  i5  720 
o.i5  693 
0.  i5  666 

9 
9 
9 

91  4i6 
91  407 
91  398 

9 

8 

7 

54 
55 
56 

9.75  75i 
9.75  769 
9.75  787 

18 
18 

9.84  36i 
9.84  388 
9.84415 

27 
27 

o.i5  639 
0.  i5  612 
0.  i5  585 

9 
9 
9 

91  389 
91  38i 
91  372 

8 
9 

6 
5 
4 

57 
58 
59 

9.75  805 
9.75  823 
9.75841 

18 
18 

9.84442 
9.84469 
9.84496 

27 
27 

27 

o.i5  558 
0.  i5  53i 
0. 1 5  5o4 

9 
9 
9 

91  363 
91  354 
91  345 

9 
9 
9 

3 
2 

I 

0 

60 

9.75859 

9.84  523 

0.  i5  477 

9 

91  336 

L.  Cos. 

d. 

L.  Cotgr. 

d. 

L.  Tang. 

I 

..  Sin. 

d. 

1 

65°.                                                      1 

PP 

.2 

•3 

28 

87 

36 

.1 

.2 

•3 

19 

18 

9 

8 

2.8 

5-6 
8.4 

2 
8 

7 

4 

I 

2.6 

5.2 

7.8 

It 

5-7 

1.8 

3-6 
5-4 

.1       0.9 
.2       1.8 

•3       2.7 

0.8 
1.6 
2.4 

•4 

•S 
.6 

II. 2 

14.0 
16.8 

10 
13 
16 

8 
5 
2 

10.4 
13.0 

15-6 

•4 

•5 

.6 

7.6 
9-5 
11.4 

7.2 
9.0 
10.8 

•4          3-6 
•5          4-5 
.6          5-4 

3-2 

4.0 
4.8 

•7 
.8 

19.6 
22.4 
2^.2 

18 
21 
24 

9 
6 

18.2 
20.8 

•33 
152 

12.6 
14.4 
16.2 

•7          6-3 
.8          7.a 
.9          8.1 

S.6 
6.4 

99 


36°. 


/ 

L.Sin.    1  d.  1 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.75869 

A 

9.84  523 

27 
26 
27 

o.i5  477 

9.91  336 

8 
9 
9 

60 

59 
58 
57 

I 

2 

3 

9.75877 
9.75895 
9.75913 

18 
18 
18 

9.84  550 
9.84576 
9.84603 

0.  i5  45o 
0.  i5  424 
0.  i5  397 

9 
9 
9 

91  328 
91  3i9 
91  3io 

4 
5 
6 

7 
8 

9 

9.75931 
9.75949 
9.75967 

9.759S5 
9.76  oo3 
9.  76  021 

- 

18 
18 
18 
[8 
18 
18 
8 
8 
8 
8 

9.84  63o 
9.84  657 
9.84  684 

9.84  7ti 
9.84738 
9.84764 

27 
27 
27 

27 
26 

27 
27 
27 
27 
27 

0.  i5  370 
o.i5  343 
o.i5  3i6 

o.i5  289 
0.  i5  262 
o.i5  236 

9 
9 
9 

9 
9 
9 

91  3oi 
91  292 

91  283 

91  274 
91  266 
91  257 

9 
9 
9 
8 

9 
9 
9 
9 
9 

56 
55 

54 

53 

52 

5i 

50 

49 
48 

47 

10 

9.76  039 

9.84  79' 

0. 1 5  209 

9 

91  248 

1 1 

12 

i3 

9.76  o57 
9.76075 
9.76  093 

9.84818 
9.84  845 
9.84872 

0.  i5  182 
o.i5  i55 
o.i5  128 

9 
9 
9 

91  239 
91  23o 
91  221 

i4 
i5 
i6 

I? 
i8 

'9 

9.76  III 
9,76  129 
9.76  i46 

9.76  164 
9.76  182 
9.76  200 

8 

7 

8 
8 
8 
8 

8 

7 
8 
8 

9.84899 
9.84  925 

9.84  952 

9.84979 

9.85  006 
9.85  o33 

26 
27 
27 
27 
27 
26 

27 
27 

27 
26 

o.i5  loi 
o.i5o75 
0. 1 5  o48 

0.  i5  021 
0.14994 
0.  i4  967 

9 
9 
9 

9 
9 
9 

91   212 

91  2o3 

91    194 

91  i85 
91  176 
91  167 

9 
9 
9 
9 
9 
9 

9 
8 

9 
9 
9 
9 
9 
9 
9 
9 

46 
45 
44 

43 
42 
4i 

40 

39 
38 
37 

20 

9.76  218 

9.85  059 

0 . 1 4  94 1 

9 

91  i58 

21 
22 
23 

9.76  236 
9.76253 
9.76  271 

9.85  086 
9.85  ii3 
9.85  i4o 

0.  i4  914 
0.14887 
0.14860 

9 
9 
9 

91  149 
91  i4i 
91  l32 

24 
25 
26 

9.76  289 
9.76307 
9.76324 

8 

7 

3 

9.85  166 
9.85  193 
9.85  220 

27 
27 

o.i4  834 
0.  i4  807 
o.i4  780 

9 
9 
9 

91  123 
91  ii4 
91  105 

36 
35 

34 

27 
28 
29 

9.76342 
9.76  36o 
9.76378 

8 
8 

9.85  247 
9.85  273 
9.85  3oo 

26 
27 
27 

0.14753 
o.i4  727 
0. 1 4  700 

9 
9 
9 

91  096 
91  087 
91  078 

33 

32 

3i 
30 

30 

9.76  395 

1 

9.85  327 

0.14673 

9 

91  069 

L.COS.      d.  1 

L.  Cotgr. 

"dT 

L.  Tang. 

L.  Sin. 

Ti 

' 

54°  30 

'. 

1 

P? 

VJ 

36 

18 

17 

9 

8 

.  I 

.3 

•3 

2.7 

5-4 
81 

2.6 
5-2 
7.8 

.1 

■3 

1.8 
3-6 
5-4 

17 

3-4 
51 

•I 

.2 
•3 

^1 

2-7 

0.8 
1.6 

2-4 

•4 
•5 
.6 

10.8 
•3  5 
16.3 

10.4 

•3° 
1S.6 

•4 

•s 

.6 

7  a 
9.0 
10.8 

6.8 
8.S 

•4 

•5 

.6 

3-6 
4  5 
5-4 

3-2 

4.0 

4.8 

•7 
.8 

iS.g 
21. 6 

24.1 

18.2 
20.8 
23.4 

■I 

12.6 

"44 
16.2 

13.6 

•7 
.8 

6.3 
7-2 
81 

5-6 
6.4 

35°  30 . 


r 

L.  Sin.       d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.76  395 

18 

9.85  327 

0 . 1 4  673 

9.91  069 

30 

•6i 

9.76  4i3 

18 

9.85  354 

26 

0.  i4  64G 

9 

91  060 

9 

29 

32 

9.76  43i 

9.85  38o 

0.  i4  620 

9 

91  o5i 

28 

33 

9-76448 

17 

9.85  407 

27 

0.  i4  593 

9 

91  042 

9 
9 
10 

27 

34 

9. 76  466 

18 

9.85434 

26 

0. 14  566 

9 

91  o33 

26 

35 

9.76484 

9.85460 

0. 1 4  54o 

9 

91  023 

25 

36 

9.76  5oi 

17 

9.85487 

27 

0. i4  5i3 

9 

91  oi4 

9 
9 

24 

37 

9.76  519 

18 

9.85  5i4 

26 

O.I4  486 

9 

91  oo5 

23 

38 

9.76537 

9.85  54o 

0.  i4  460 

9 

90  996 

22 

39 

9.76  554 

17 

18 

9.85  567 

27 
27 

26 
27 

0.14433 

9 

90987 

9 

21 

20 

40 

9. 76  572 

9.85  594 

0. 1 4  4o6 

9 

90978 

4i 

9.76  590 

9.85  620 

o.i4  38o 

9 

90  969 

9 

'9 

42 

9.76607 

9-85  64? 

0.14353 

9 

90  960 

18 

43 

9.76625 

9.85  674 

27 
26 

0.  i4  326 

9 

90  95 1 

9 
9 
9 

'7 

44 

9.76  642 

18 

9.85  700 

27 

0.  i4  3oo 

9 

90  942 

16 

45 

9.76  660 

9.85  727 

0.  i4  273 

9 

90  933 

i5 

46 

9.76677 

17 

9.85  754 

27 

o.i4  246 

9 

90  924 

i4 

47 

9.76695 

9-85  780 

27 

0.  i4  220 

9 

90915 

9 

i3 

48 

9.76  712 

18 

9.85  807 

0.  i4  193 

9 

90  906 

12 

49 

9.76  73o 

9.85  834 

27 

0.  i4  166 

9 

90  896 

9 

9 
9 

1 1 

50 

9.76  747 

'7 
18 

9.85  860 

27 
26 

0. 1 4  i4o 

9 

90887 

10 

9 

5i 

9.76  765 

9.85887 

o.i4  ii3 

9 

90  878 

52 

9.76  7S2 

9.85913 

0.  i4  087 

9 

90  869 

8 

53 

9.76  800 

'7 

9.85  940 

27 
27 

0. 1 4  060 

9 

90  860 

9 

7 

54 

9.76  817 

18 

9.85  967 

26 

o.i4  o33 

9 

90  85i 

9 

6 

55 

9.76835 

9.85  993 

0. 1 4  007 

9 

90  842 

5 

56 

9.76  852 

»7 
18 

9.86  020 

27 
26 

0.  i3  980 

9 

90  832 

9 

4 

57 

9.76  870 

9.86046 

o.i3  954 

9 

90823 

9 

3 

58 

9.76  887 

9.86  073 

0.  i3  927 

9 

90  8i4 

2 

59 

9.76  904 

17 

9.86  100 

26 

0.  i3  900 

9 

90  805 

9 

I 

60 

9.76  Q22 

9.86  126 

o.i3  874 

9 

90  796 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

r 

54°.                                                      1 

PP 

.1 

27 

26 

z8 

17 

.1 

10 

9 

2.7 

2.6 

.1 

1.8 

1-7 

I.O 

0.9 

.2 

5-4 

S.2 

.2 

3-6 

3-4 

.2 

2.0 

1.8 

•3 

8.1 

7.8 

•  3 

5-4 

5-' 

•3 

3.0 

2-7 

•4 

10.8 

10.4 

•4 

7.2 

6.8 

•4 

4.0 

3-6 

■5 

'3-5 

13.0 

•5 

9.0 

8.5 

•5 

50 

4-5 

.6 

16.2 

15.6 

.6 

10.8 

10.2 

.6 

6.0 

5-4 

•7 

18.9 

18.2 

.7 

ia.6 

11.9 

.7 

7.0 

6-3 

.8 

21.6 

20.8 

.8 

14.4 

13.6 

.8 

8.0 

7-2 

•Q     1       24.3       1       23.4 

.Q           16.2 

^^ 

.9    1        9.0       1        8.1      1 

36°. 


' 

L.  Sin. 

d. 

L.  Tang.  |  d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.76  922 

9.86  126 

27 
26 

0.13874 

9.90  796 

60 

I 

9 

.76939 

18 

9.86  i53 

O.I3847 

9.90787 

9 

59 

2 

9 

.76967 

9.86  179 

27 
26 

O.I382I 

9.90777 

58 

3 

9 

.76974 

17 
18 

9.86  206 

o.i3  794 

9.90  768 

^7 

4 

9 

.76991 

9.86232 

27 

o.i3  768 

9.90769 

56 

5 

9 

.77009 

9.86259 

26 

o.i3  741 

9.90750 

55 

6 

9 

.77  026 

•7 
17 

9.86285 

27 

o.i3  715 

9.90  741 

9 
10 

54 

7 

9 

.77043 

18 

9.86  312 

26 

0. 13  688 

9.90731 

9 

53 

8 

9 

.77  061 

9.86  338 

0.1 3  662 

9.90  722 

52 

9 

9 

.77078 

'7 

9.86  365 

27 

0. 13635 

9.90  713 

y 

5i 

'0.1 3  608 

10 

9 

77095 

17 
18 

9.86  392 

26 

9.90  704 

10 
9 

50 

II 

9 

77  112 

9.86418 

0. 13582 

9.90694 

49 

12 

9 

77  i3o 

9.86445 

26 

0. 13555 

9.90  685 

48 

i3 

9 

77  i47 

17 

9.86471 

o.i3  529 

9.90  676 

9 

47 

17 

27 

9 

i4 

9 

77  i64 

9.86498 

26 

0.1 3  5o2 

9.90  667 

4b 

i5 

9 

77  181 

9.86  524 

0.13476 

9.90  667 

45 

i6 

9 

77  199 

18 

9.86  551 

27 

0.13449 

9.90  648 

9 

44 

I? 

9 

77  216 

•7 

9.86677 

26 

0.  i3  423 

9.90  639 

9 

43 

i8 

9 

77233 

9.86603 

o.i3  397 

9.90  63o 

42 

'9 

9 

77  25o 

■7 

9.86  63o 

27 
26 

o.i3  370 

9.90  620 

4i 

20 

9 

77  268 

9.86  656 

0.1 3  344 

9.90  61 1 

9 

40 

21 

9 

77285 

9.86  683 

26 

o.i3  317 

9.90  602 

9 

39 

22 

9 

77  3o2 

9.86  709 

o.i3  291 

9.90  692 

38 

23 

9 

77319 

'7 

9.86  736 

27 
36 

o.i3  264 

9.90683 

9 

37 

24 

9 

77  336 

'7 

9.86  762 

27 

O.I3238 

9.90  674 

9 

36 

25 

9 

77353 

9.86789 

o.i3  211 

9.90665 

36 

26 

9 

77370 

'7 
>7 

9.86815 

27 

o.i3  185 

9.90  555 

9 

34 

27 

9 

77387 

18 

9.86  842 

36 

o.i3  i58 

9.90  546 

9 

33 

28 

9 

77405 

9.86868 

26 

o.i3  i32 

9.90  537 

32 

29 

9 

77422 

'7 

9.86894 

0. 1 3  106 

9.90  627 

3i 

"7 

27 

9 

80 

9 

77439 

9.86  921 

0.  i3  079 

9.90  5i8 

30 

L.Cos.    1   d.  1 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

53°  30 

', 

PP 

.1 

37 

36 

.1 

18 

17 

.1 

10 

9 

2.7 

3.6 

1.8 

'•7 

I.O 

T.t 

.3 

5* 

5-5 

.2 

3.6 

3-4 

.3 

2.0 

•3 

8.1 

7.8 

•3 

5-4 

S.i 

•3 

3.0 

3.7 

•4 

10.8 

10.4 

•4 

7.2 

6.8 

•4 

4.0 

3.6 

■5 

'3-5 

13.0 

■  5 

90 

8.5 

•s 

5-0 

4-5 

.6 

16.2 

15.6 

.6 

xo,8 

I0.3 

.6 

6.0 

5.4 

•7 

18.9 

18.2 

•7 

12.6 

11.9 

•7 

7.0 

6.3 

.8 

21.6 

20.8 

.8 

14-4 

13.6 

.8 

8.0 

7.2 

16.2 

^^ 

^^ 

8.1 

102 


36 

°30 

• 

' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.Cos.     d. 

30 

9.77439 

9.S6  921 

26 
27 

0. 1 3  079 

9.90  5i8 

9 

30 

3i 

9 

77  456 

9.86  947 

0.  i3  o53 

9 

90  509 

29 

32 

9 

77473 

9.86974 

26 

0.  i3  026 

9 

90  499 

28 

33 

9 

77490 

g.87  000 

27 

0.  i3  000 

9 

90  490 

9 
10 

27 

34 

9 

77507 

9.87  027 

26 

0. 12  973 

9 

90  480 

9 

26 

35 

9 

77  524 

9.87  o53 

26 

0.12  947 

9 

90471 

25 

36 

9 

77541 

9.87079 

27 

0. 12  921 

9 

90  462 

10 

24 

3? 

9 

77b58 

9.87  106 

26 

0.12  894 

9 

90452 

9 

23 

38 

9 

77575 

9.87  l32 

26 

0.12868 

9 

90443 

22 

39 

9 

77592 

9.87  i58 

0. 12  842 

9 

90434 

9 

21 

40 

9 

77  609 

9.87  185 

26 

0.12815 

9 

90  424 

9 

20 

4i 

9 

77626 

9.87  211 

0. 12  789 

9 

90415 

19 

42 

9 

77643 

9.87238 

26 

0.12  762 

9 

90  4o5 

18 

43 

9 

77  660 

9.87  264 

0. 12  736 

9 

90  396 

9 

17 

26 

10 

44 

9 

77677 

9.87  290 

27 
26 

0.12  710 

9 

90  386 

9 

16 

45 

9 

77  694 

9.87817 

0.12  683 

9 

90377 

i5 

46 

9 

77711 

9.87343 

0.12  657 

9 

90  368 

9 

i4 

26 

10 

47 

9 

77728 

9.87  369 

27 
26 
26 

27 

0.12  63i 

9 

90  358 

9 

i3 

48 

9 

77  744 

9.87  396 

0.12  6o4 

9 

90  349 

12 

49 

9 

77761 

9.87  422 

0. 12  578 

9 

90  339 

9 
10 

II 

50 

9 

77  778 

9.87448 

0. 12  552 

9 

90  33o 

10 

bi 

9 

77795 

9.87475 

26 

0. 12  525 

9 

90  320 

9 

9 

52 

9 

77812 

9.87  5oi 

0. 12  499 

9 

.90  3ii 

8 

53 

9 

77  829 

9.87  627 

0. 12  473 

9 

.90  3oi 

7 

54 

9 

77846 

9.87554 

27 

26 

0.12  446 

9 

.90  292 

9 

6 

55 

9 

77  862 

9.87  58o 

26 

0. 12  420 

9 

.90  282 

5 

56 

9 

77879 

9.87  606 

0.12  394 

9 

.90  273 

9 

4 

27 

10 

^1 

9 

77  896 

9.87633 

26 

0.12  367 

9 

.90  263 

9 

3 

58 

9 

77913 

9.87659 

0. 12  34i 

9 

.90  254 

2 

59 

9 

.77930 

9.87685 

26 

0. 12  3i5 

9 

.90  244 

I 

60 

9 

.77  946 

9.87  711 

0.12  289 

9 

.90235 

0 

L.  Cos. 

d. 

L.  Cotg.      d. 

L.  Tang. 

L.  Sin. 

d. 

' 

53°.                                                        1 

PP 

127 

26 

«7 

16 

10 

9 

.1 

■2.7 

2.6 

.1 

>-7 

1.6 

.1 

I.O 

0.9 

.2 

5-4 

5.2 

.2 

3-4 

3? 

2.0 

1.8 

•3 

8.1 

7.8 

•3 

51 

4.8 

30 

2.7 

■4 

10.8 

10.4 

.4 

6.8 

6.4 

4.0 

3-6 

•  5 

«3-5 

13.0 

.5 

8.5 

8.0 

5.0 

4  5 

.6 

16.2 

15.6 

.6 

10.2 

96 

.( 

6.0 

S-4 

•7 

i8.g 

18.2 

•7 

II  9 

II. 2 

7.0 

6.3 

.8 

21.6 

20.8 

.8 

136 

12.8 

. 

8.0 

7.2 

24_^_ 

_2^ 

^1^ 

. 

L. 

81 

io3 


37°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.    !d.| 

0 

9.77946 

9.87711 

0.12  289 

9.90  235 

60 

I 

9.77963 

17 

9.87738 

36 

0.12  262 

9.90  225 

9 

59 

2 

9.77980 

»7 

16 

9.87  764 

26 

0.12  236 

9.90  216 

58 

3 

9.77997 

9.87790 

27 

0.12  210 

9.90  206 

9 

^7 

4 

9.78  oi3 

17 

9.87817 

36 

0.12  i83 

9.90197 

10 

56 

5 

9.78  o3o 

9.87843 

0.12  157 

9.90  187 

55 

6 

9.78  047 

16 

9.87869 

26 

0.12  i3i 

9.90  178 

54 

7 

9.78  o63 

17 

9.87895 

0.12  105 

9.90  168 

9 

53 

8 

9.78  080 

9.87  922 

26 
26 

0.12  078 

9.90  i59 

52 

9 

9.78097 

9.87948 

0. 12  o52 

9.90  149 

5i 

10 

9.78  1 13 

17 
"7 

9.87974 

0.12  026 

9.90  139 

50 

II 

9.78  i3o 

9.88  000 

27 

0.12  000 

9.90  i3o 

49 

12 

9.78  147 

16 

9.88  027 

o.ii  973 

9.90  120 

48 

i3 

9.78  i63 

17 

9.88053 

26 

0.1 1  947 

9.90  III 

9 
10 

47 

i4 

9.78  180 

17 

9.88  079 

26 

O.II  921 

9.90  lOI 

4b 

i5 

9.78  197 

16 

9.88  io5 

O.II  895 

g. 90  091 

45 

i6 

9.78213 

17 

16 

9.88  i3i 

O.II  869 

9.90  082 

y 

44 

I? 

9.78  23o 

9.88  i58 

*7 
26 

O.II  842 

9.90  072 

9 

43 

i8 

9.78  246 

9.88  184 

O.II  816 

9.90  o63 

42 

'9 

9.78  263 

9.88  210 

O.II  790 

9.90  o53 

4i 

20 

9.78  280 

16 

9.88236 

26 

O.II  764 

9.90  o43 

40 

21 

9.78  296 

17 

9.88  262 

O.II  738 

9.90  o34 

9 

39 

22 

9.78313 

16 

9.88289 

26 

O.II  711 

9.90  024 

38 

23 

9.78  329 

9.88315 

O.II  685 

9.90  oi4 

37 

«7 

26 

9 

24 

9.78346 

16 

9.88341 

26 

O.II  659 

9.90005 

36 

25 

9.78  362 

9.88367 

26 

O.II  633 

9.89995 

35 

26 

9.78379 

16 

9.88393 

O.II  607 

9.89  985 

34 

27 

9.78  395 

17 

9.88420 

27 
26 

O.II  58o 

9.89976 

9 

33 

28 

9.78  4l2 

9.88  446 

O.II  554 

9.89  966 

32 

29 

9.78428 

9.88472 

26 

O.II  528 

9.89  956 

3i 

30 

9.78445 

9.88  498 

O.II  5oa 

9.89947 

9 

30 

L.  Cos. 

d. 

L.  Cotg.  1  d. 

L.  Tang. 

L.  Sin.     d. 

' 

52°  3C 

1 

PP 

.1 

87 

3« 

17 

16 

.1 

10 

9 

2.7 

2.6 

.1 

1-7 

1.6 

I.O 

0.9 

.2 

5-4 

5-2 

.2 

3-4 

3-2 

.3 

2.0 

1.8 

•3 

8.1 

7.8 

•3 

51 

4.8 

•3 

3.0 

27 

•4 

10.8 

10.4 

•4 

6.8 

6.4 

•4 

4.0 

3.6 

•5 

'3-5 

130 

.5 

8-5 

8.0 

•5 

50 

45 

.6 

16.3 

15-6 

.6 

10.2 

9.6 

.6 

6.0 

5-4 

.7 

18.9 

18.2 

•  7 

11.9 

II. 2 

•  7 

7.0 

63 

.8 

21.6 

20.8 

.8 

136 

12.8 

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9.0      1       8. 1      1 

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37 

°30 

'. 

' 

L.  Sin. 

d.    I 

..  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.78445 

5.88  498 

26 
26 

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9.89947 

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9.89  722 

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0 

L.  Cos. 

d.     L.  Cotgr. 

d. 

L,  Tang. 

L.  Sin. 

d. 

' 

52°.                                                      1 

PP 

27 

36 

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17 

16 

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10 

9 

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2-7 

2.6 

1-7 

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38°. 


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L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.78  934 

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17 
16 
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26 
26 
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9 
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16 
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16 
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10 
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16 
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26 
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9 
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16 
16 
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16 
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16 
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26 
26 
26 

26 
26 
26 
26 

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9 
9 
9 

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89475 
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10 
10 
10 
10 
10 
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42 
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39 
38 
37 

20 

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9 

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9 
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25 

26 

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9 
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9 
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10 
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36 
35 
34 

27 
28 
29 

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16 
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9 
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10 

33 

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30 

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90  061 

0.09  939 

9 

89354 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin.     d. 

' 

51°  30.                                                1 

PP 

.1 

.2 

•3 

36 

17 

16 

.1 
.2 

•3 

15 

IX 

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10 

9 

2.6 

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6.8 
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8.0 
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3.6 

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12.8 

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12 

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7.0 
8.0 

6.3 

7-2 

8.1 

106 


38 

°30 

• 

/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg.      1 

..  Cos. 

d. 

30 

9.79415 

16 
16 
16 

9.90  061 

25 
26 
26 
26 

0.09939     9 

89  354 

10 
10 
10 

30 

3i 

32 

33 

9 
9 
9 

79431 
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79  463 

9.90  086 
9.90  112 
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0.09914     9 
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29 
28 
27 

34 
35 
36 

9 

9 
9 

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16 
16 
16 

9.90  164 
9.90  190 
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26 
26 
26 

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0.09810     9 
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89  8i4 
89  8o4 
89  294 

10 
10 

26 

25 

24 

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38 
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9 
9 
9 

79  526 
79  542 
79  558 

16 
16 

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16 
16 
16 
15 

9.90  242 
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26 
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26 

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10 
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10 

23 
22 
21 

40 

9 

79  573 

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9 
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44 
45 
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16 
16 
16 

9.90  423 
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10 
10 

16 
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9 
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16 
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15 
16 
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9.90  553 

26 
26 

25 

26 
26 
26 

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10 

10 

10 
10 
10 

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12 
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52 

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9 

8 

7 

54 
55 
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26 
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6 
6 
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57 
58 
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26 
26 

26 

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10 
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3 
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L.  Cos. 

d. 

L.  Cotg.  1  d. 

L.  Tang.      1 

L.  Sin. 

d. 

/ 

51°. 

PP 

.1 

.2 

•3 

26 

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16 

15 

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! 
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II 

10 

2.6 

5-2 

7.8 

2-5 

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6.4 
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4.4 

55 
6.6 

4.0 

50 
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12.8 

10.5 
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r 

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7-7 
8.8 

7.0 
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107 


39°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.    i  d. 

0 

9.79887 

16 

9.90  837 

26 
26 

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60 

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30 

L.  Cos.    1  d.  1 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin.     d7| 

' 

50°  30 

►'. 

1 

PP 

.1 

36 

as 

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16 

15 

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10 

2.6 

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1-5 

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77 

7.0 

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20.8 

20.0 

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8.8 

8.0 

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108 


39°  30 

', 

' 

L,  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.80  35i 

9.91  6io 

26 
26 
26 

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26 
26 
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32 

33 

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29 

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80  610 
80625 
8o64i 

IS 

16 

9.92  o48 
9.92  073 
9.92099 

25 
26 
26 

25 

26 
26 

0.07  952 
0.07  927 
0.07  901 

9 
9 
9 

.88  563 
.88  552 
.88  542 

i3 
12 
1 1 

50 

9 

80  656 

IS 

9.92  125 

0.07  875 

9 

.88  531 

10 

5i 

52 

53 

9 
9 
9 

80671 

80686 

.80  701 

15 

IS 

15 

9.92  i5o 
9.92  176 
9.92  202 

0.07  850 
0.07  824 
0.07  798 

9 
9 
9 

.88  521 
.88  5io 
.88  499 

9 

8 

7 

54 
55 
56 

9 
9 
9 

.80716 
.80731 
.80746 

IS 

15 

IS 

16 

IS 
IS 

9.92  227 
9.92  253 
9.92279 

*5 
26 
26 

0.07  773 
0.07  747 
0.07  721 

9 
9 
9 

.88489 
.88478 
.88  468 

6 

5 
4 

5? 
58 
59 

9 
9 
9 

.80  762 
.80777 
.80  792 

9.92  3o4 
9.92  33o 
9.92  356 

26 
26 
25 

0.07  696 
0.07  670 
0.07  644 

9 
9 
9 

.88457 

.88  447 
.88  436 

10 

3 
2 
I 

60 

9 

.80807 

IS     - 

9.92  38i 

0.07  619 

9 

.88  425 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

50°. 

1 

PP 

.1 

.2 

•3 

26 

25 

.t 
.2 

•3 

16 

15 

JI 

10 

2.6 
5-2 

7.8 

2S 

7-5 

1.6 

3-2 

4.8 

30 

4S 

. 

X.I 

2.3 

3-3 

I.O 

2.0 

30 

•4 
•5 
.6 

10.4 
13.0 

156 

10. 0 
12.5 
150 

•4 
•s 

.6 

6.4 
8.0 
9.6 

6.0 
7-5 
9.0 

. 

4-4 

1:1 

4.0 
50 
6.0 

•7 
.8 

18.  2 

2a8 

175 
20.0 

.7 
.8 

11.2 
12.8 

10.5 
12.0 

! 

7-7 
8.8 

7.0 
8.0 

•9           23-4       1 

22.5 

109 


40°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

0 

9.80  807 

15 

9.92  38i 

26 
26 

0.07  619 

9.88425 

lO 

60 

I 

9.80822 

9.92  407 

0.07  593 

9 

88415 

59 

2 

9.80837 

9.92433 

0.07  567 

9 

88  4o4 

58 

3 

9.80  852 

'S 

9.92  458 

26 

0.07  542 

9 

88394 

57 

4 

9.80867 

15 

9.92484 

26 

0.07  5i6 

9- 

88  383 

56 

5 

9.80882 

9.92  5io 

0.07  490 

9- 

88372 

55 

6 

9.80  897 

'5 

9.92  535 

25 
26 

0.07  465 

9- 

88  362 

54 

7 

9.80  912 

9.92  56i 

26 

0.07  439 

9- 

88  35i 

53 

8 

9.80  927 

9.92  587 

0.07  4i3 

9- 

88  340 

52 

9 

9.80  942 

9.92  612 

25 
26 

25 
26 

0.07  388 

9- 

88  33o 

II 
II 

5i 

10 

9.80  957 

9.92  638 

0.07  362 

9- 

88  3i9 

50 

II 

9.80  972 

9.92  663 

0.07  337 

9- 

88  3o8 

49 

12 

i3 

9.80  987 

9.81  002 

''' 

9.92  689 
9.92715 

26 

25 

0.07  3i  I 
0.07  285 

9- 
9- 

88  298 
88  287 

II 
II 

48 
47 

i4 

9.81  017 

9.92  740 

26 

0.07  260 

9- 

88276 

4b 

i5 

9.81  o32 

9.92  766 

0.07  234 

9- 

88266 

40 

i6 

9.81  047 

9.92792 

25 

0.07  208 

9- 

88255 

II 

44 

17 

9.81  061 

9.92  817 

26 

0.07  i83 

9- 

88244 

43 

i8 

9.81  076 

9.92  843 

0.07  157 

9- 

88  234 

42 

'9 

9.81  091 

9.92868 

25 

0.07  l32 

9- 

88223 

4i 

20 

9.81  106 

9.92  894 

26 

0.07  106 

9- 

88  212 

II 

40 

21 

9.81    121 

9.92  920 

0.07  080 

9- 

88  201 

39 

22 

9.81  i36 

* 

9.92  945 

0.07  055 

9- 

88  191 

38 

23 

9.81  i5i 

9.92971 

25 

0.07  029 

9- 

88  180 

II 

37 

24 

9.81  166 

9.92996 

26 

0 .  07  oo4 

9- 

88  169 

36 

25 

9.81  180 

9.93  022 

26 

25 

26 

0.06  978 

9- 

88  i58 

35 

26 

9.81  195 

9.93048 

0.06  952 

9- 

88  i48 

10 

34 

27 

9.81  210 

9.93  073 

0.06  927 

9- 

88  137 

33 

28 

9.81  225 

9.93099 

0.06  901 

9- 

88  126 

32 

29 

9.81  240 

9.93  124 

26 

0.06  876 

9- 

88  ii5 

10 

3i 

30 

9.81  254 

9.93  150 

0.06  85o 

9- 

88  105 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin.    |d.| 

' 

49°  30 

'. 

1 

PP 

.1 

36 

as 

«5 

«4 

.1 

II 

10 

2.6 

2-5 

.1 

i-S 

'•4 

I.I 

I.O 

■2 

5-2 

5.0 

.3 

30 

2.8 

.2 

2.2 

2.0 

•3 

7-8 

7-5 

•3 

4-S 

4.2 

•3 

3-3 

30 

•4 

10.4 

lO.O 

•4 

6.0 

56 

•4 

4-4 

40 

•5 

>30 

•2.5 

•s 

7-5 

7.0 

•  5 

55 

5-0 

.6 

15.6 

I5-0 

.6 

9.0 

8.4 

.6 

6.6 

6.0 

•7 

18.3 

>7-5 

•7 

lo.s 

9.8 

•7 

7-7 

7.0 

.8 

20.8 

20.0 

.8 

12.0 

II. 2 

.8 

8.8 

8.0 

12.6 

^^^ 

•9    1        9-9       1        90      1 

40°  3C 

', 

/ 

L.  Sin.       d. 

L.  Tang,  i   d. 

L.  Cotg. 

L.  Cos.     d. 

30 

9.81  254 

'5 

M 

'5 

9.93  150 

25 
26 
26 

25 

26 

25 

0.06  b5o 

9.88  105 

10 

30 

3i 

32 

33 

34 
35 
36 

9 
9 
9 

9 
9 
9 

.8i  269 
.81  284 
.81  299 

.81  3i4 

.81  328 
.81  343 

9.93  175 
9.93  201 
9.93  227 

9.93  252 
9.93  278 

9.93  3o3 

0.06  825 
0.06  799 
0.06  773 

0.06  748 
0.06  722 
0.06  697 

9 
9 
9 

9 
9 
9 

88  094 
88o83 
88  072 

88061 
88o5i 
88o4o 

29 
28 
27 

26 

25 

24 

37 

38 
39 

9 
9 
9 

.81  358 

.81  372 
.81  387 

M 

M 

'5 

9.93  329 
9.93  354 
9.93  38o 

25 
26 
26 

25 
26 

25 

0.06  671 
0.06  646 
0.06  620 

9 

9- 

9- 

88  029 
88018 
88007 

23 
22 
21 

20 

40 

9 

.81  402 

9.93  4o6 

0.06  594 

9- 

87  996 

4i 
42 
43 

9 
9 
9 

.81  417 
.81  43i 
.81  446 

9.93  43i 
9.93  457 
9.93  482 

0.06  569 
0.06  543 
o.o65i8 

9- 
9- 
9- 

87985 

87975 
87964 

19 

18 

17 

44 
45 
46 

9 
9 
9 

.81  46i 
.81  475 
.81  490 

14 
15 

9.93  5o8 
9.93  533 
9.93  559 

25 
26 

0.06  492 
0.06  467 
0.06  44i 

9- 
9- 
9- 

87953 
87  942 
87  931 

16 
i5 

i4 

47 
48 

49 

9 
9 
9 

.81  505 

.81  519 

81  534 

14 
15 
'5 

'4 
'5 
14 

9.93584 
9.93  610 
9.93  636 

26 
26 

25 

26 

25 

26 

0.06  4i6 
0.06  390 
0.06  364 

9- 
9- 
9- 

87  920 
87  909 

87  898 

i3 
12 
II 

10 

9 

8 

7 

50 

9 

81  549 

9.93  661 

0.06  339 

9- 

87887 

5i 

52 

53 

9 
9 
9 

81  563 
81  578 
81  592 

9.93  687 
9.93  712 
9.93738 

0.06  3i3 
0.06  288 
0.06  262 

9- 
9- 
9- 

87877 
87866 
87855 

54 
55 
56 

9 
9 
9 

81  607 
81  622 
81  636 

'5 
"4 

9.93  763 
9.93  789 
9.93814 

26 

25 

26 
25 
26 

25 

0.06  237 
0.06  21 1 
0.06  186 

9- 
9- 
9- 

87844 
87833 
87  822 

6 

5 
4 

57 
58 
59 

9 
9 
9 

81  65i 
81  665 
81  680 

14 
15 
14 

9.93  84o 
9.93  865 
9.93  891 

0.06  160 
0.06  i35 
0.06  109 

9- 
9- 
9- 

87811 
87  800 
87789 

3 
2 

I 

0 

60 

9 

81  694 

9.93  916 

0.06  084 

9- 

87778 

L.  Cos.       d.  1 

L.  Cotg.  1   d.  1 

L.  Tang. 

L.  Sin.    i 

d. 

f 

49°.                                                        1 

PP 

.1 
.2 

•3 

26 

25 

.2 
•3 

«5 

14 

.1 
.2 

•3 

II 

10 

2.6 

5.2 

7.8 

2-5 

7-5 

1-5 
3.0 

45 

1.4 
2.8 
4.2 

I.I 

2.2 

3-3 

1.0 
2.0 

30 

•4 
•s 

.6 

10.4 
13.0 

15.6 

10. 0 

12  5 

15.0 

•4 

•  5 

.6 

6.0 
75 
9.0 

5-6 

1° 
8.4 

•4 

•5 

.6 

4  4 
5-5 
6.6 

4.0 
50 
6.0 

•7 
.8 

•Q 

l8.2 

20.8 

23.4 

•7-5 
20.0 

•7 

.8 

•  0 

10.5 
12.0 

.3.5 

9.8 
II. 2 

12.6 

.^ 

11 

7.0 
8.0 

41°. 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

i. 

0 

9.81  694 

-      IS 

14 

IS 

9.93  916 

26 

25 
26 

0.06  oy4 

9.87778 

60 

.  59 

58 
'     57 

I 

2 

3 

9.81  709 
9.81  723 
9.81  738 

9.93942 
9.93967 
9.93993 

0.06  o58 
0.06  o33 
0.06  007 

9.87767 
9.87756 
9.87745 

4 
5 
6 

9.81  752 
9.81  767 
9.81  781 

15 
14 

9.94  018 
9.94044 
9.94069 

26 

2S 

26 

o.o5  982 
0.05956 
o.o5  931 

9.87734 
9.87  723 
9.87  712 

56 

55 

'     54 

7 
8 

9 

9.81  796 
9.81  810 
9.81  825 

14 

IS 

1     la 

9.94095 
9.94  120 
9.94  i46 

25 
26 

25 

26 

25 

26 

25 

o.o5  905 
o.o5  880 
0.05  854 

9.87701     J 
9.87  690 
9.87679    ■ 

53 

52 

'      5i 
50 

I     49 

.     48 

r       ^^7 

10 

9.81  839 

'5 
14 
14 
IS 

9.94  17' 

o.o5  829 

9.87668 

II 

12 

i3 

9.81  854 
9.81  868 
9.81  882 

9.94  197 
9.94  222 
9.94248 

o.o5  8o3 
o.o5  778 
o.o5  752 

9.87657     , 
9.87646 
9.87635    ^ 

i4 
i5 
i6 

9. 81897 
9.81  911 
9.81  926 

14 

15 

9.94273 
9-94299 
9.94324 

26 

25 

26 

o.o5  727 
o.o5  701 
o.o5  676 

9.87624    , 
9.87613 
9.87601     ' 

,     46 

45 
44 

i8 
'9 

9.81  940 
9.81955 
9.81  969 

15 
14 
14 

15 
14 
14 
15 
14 
14 
IS 

14 
14 
14 

9-94  350 
9.94375 
9.94401 

25 
26 

25 
26 
25 
26 

o.o5  65o 
o.o5  625 
o.o5  599 

9.87590    , 

9.87579 

9.87568 

I     43 

42 

'     4i 

40 

I     39 

38 

;  37 

20 

9.81  983 

9.94426 

o.o5  574 

9.87557 

21 
22 
23 

9.81998 
9.82  012 
9.82  026 

9.94452 

9.94477 
9.94  5o3 

0.05  548 
o.o5  523 
o.o5  497 

9.87546     , 
9.87535     ^ 
9.87  524 

24 
25 
26 

9.82  o4 1 
9.82  055 
9.82  069 

9.94  528 
9.94  554 
9.94579 

26 
25 

o.o5  472 
o.o5  446 
o.o5  421 

9.87513     , 
9.87  5oi 
9.87490 

J      36 

35 

'     34 

27 
28 
29 

9.82084 
9.82  098 
9.82  112 

9.94604 
9.94  63o 
9.94  655 

26 

25 

26 

o.o5  396 
o.o5  370 
o.o5  345 

9.87479     , 
9.87468 
9-87457     ' 

,     33 

32 

'     3i 
30 

30 

9.82  126 

9.94  681 

o.o5  319 

9.87446 

L.Cos.      d.  1 

L.  Cotgr. 

d. 

L.  Tang. 

L.  Sin.     d 

. 

48°  30 

. 

PP 

.1 
.2 

•3 

36 

as 

»5 

M 

.1 

.2 

•3 

la 

11 

2.6 
7.8 

2  S 
50 
7-5 

.1 

.2 

•3 

i-S 
30 
4  5 

1-4 

2.8 
4-2 

1.2 

2.4 
3-6 

I.I 

2.2 

3-3 

•4 
•5 

.6 

10.4 
13.0 

15-6 

10.0 
12.5 
150 

•4 

6.0 
7S 
9.0 

5-6 

•4 
•5 
.6 

4.8 
6.0 

7-2 

4-4 

5-5 
6.6 

•7 
.8 

18.2 
20.8 

175 
20.0 

/■I 

10.  S 
12.0 

9.8 

II. 2 
12.6 

•7 
.8 

8.4 
9.6 
10.8 

7-7 
8.8 

41°  30 . 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.S2  126 

15 

9.94  681 

25 

26 

o.o5  3i9 

9.87446 

12 

30 

29 

3i 

9.82  i4i 

9.94  706 

o.o5  294 

9.87434 

32 

9.82  155 

9.94  732 

o.o5  268 

9.87  428 

28 

33 

9.82  169 

9.94757 

25 
26 

o.o5  243 

9.87  4l2 

II 

27 

34 

9.82  i84 

9.94783 

o.o5  217 

9.87  4oi 

26 

35 

9.82  198 

9.94808 

o.o5  192 

9.87  890 

25 

36 

9.82  212 

9.94834 

o.o5  166 

9.87  878 

24 

3? 

9.82  226 

9.94  859 

o.o5  i4i 

9.87  867 

28 

38 

9.82  240 

9.94884 

o.o5  116 

9.87  356 

22 

39 

9.82  255 

14 
14 

'4 

9.94  910 

25 

26 
25 

o.o5  090 

9.87845 

II 

12 

21 

40 

9.82  269 

9.94  935 

o.o5  065 

9.87  834 

20 

4i 

9.82283 

9.94  961 

o.o5  089 

9.87  822 

19 

42 

g.82  297 

9.94  986 

26 

o.o5  oi4 

9.87  3ii 

18 

43 

9.82  3ii 

'5 

9.95  012 

25 

0.04988 

9.87  3oo 

12 

17 

44 

9.82  326 

14 

9.95  087 

o.o4  963 

9.87288 

16 

45 

9.82  34o 

9.95  062 

26 

o.o4  988 

9.87277 

i5 

46 

9.82  354 

9.95  088 

o.o4  912 

g.87  266 

i4 

47 

9.82  368 

9.95  1 13 

25 
26 

o.o4  887 

9.87  255 

18 

48 

9.82  382 

9.95  i3g 

0.04861 

9.87243 

12 

49 

9.82  396 

14 

9.95  164 

25 

26 
25 

25 

o.o4  836 

9.87  282 

1 1 
10 

50 

9.82  4io 

9.95  190 

0 . o4  8 1 0 

9.87  221 

5i 

9.82  424 

15 

9.95  2l5 

o.o4  785 

9.87  209 

9 

02 

9.82439 

9.95  24o 

26 

o.o4  760 

9.87  198 

8 

53 

9.82453 

g.95  266 

o.o4  734 

9.87187 

7 

54 

9.82  467 

9.95  291 

26 

o.o4  709 

9.87  175 

6 

55 

9.82481 

9.95  3i7 

o.o4  683 

9.87  164 

5 

56 

9.82  495 

9.95  342 

26 

0.04  658 

9.87  i58 

4 

57 

9.82  509 

9.95  368 

25 

o.o4  682 

9.87  i4i 

3 

58 

9.82  523 

9.95  893 

o.o4  607 

9.87  180 

2 

59 

9.82  537 

14 
14 

9.95  4i8 

26 

o.o4  582 

9.87  119 

12 

I 
0 

60 

9.82  55i 

9.95  444 

o.o4  556 

9.87  107 

L.  Cos.      d.  1  L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

48°.                                                      1 

PP 

.1 

26 

as 

15 

1-5 

14 
1.4 

.1 

13 

II 

2.6 

2-5 

I 

1.2 

I.I 

.2 

52 

50 

2 

30 

2.8 

.2 

2.4 

2.2 

3 

7.8 

7-5 

•3 

4-S 

4-2 

•3 

3.6 

3-3 

4 

10.4 

lO.O 

•4 

6.0 

5.6 

•4 

4.8 

4.4 

■  5 

13.0 

12.5 

•5 

7-5 

7.0 

■5 

6.0 

5-5 

.6 

.5-6 

15.0 

.6 

9.0 

8.4 

.6 

7.2 

6.6 

■7 

18.2 

'7-5 

•7 

10.  s 

9.8 

•7 

8.4 

7  7 

8 

20.8 

20.0 

.8 

12.0 

II. 2 

.8 

9.6 

8.8 

22.5 

12.6 

^^ 

.^2. 

io8 

ii3 


42°. 


0 

L.  Sin. 

1  d. 

L.Tang.  1  d. 

L.  Cotg. 

L.Cos.    |d.|       1 

9.82  55i 

14 
14 
14 

9.95  444 

o.o4  556 

9.87  107 

II 
II 
12 

60 

I 

2 

3 

9.82  565 
9.82  579 
9.82593 

9.95  46y 

9.95495 
9.95  520 

26 
25 

o.o4  53i 
o.o4  5o5 
o.o4  48o 

9.87  096 
9.87085 
9.87073 

59 

58 
57 

4 
5 
6 

9.82  607 
9.82  621 
9.82635 

M 
»4 

9.95  545 
9.95  571 
9.95  596 

26 

25 

26 

o.o4  455 
o.o4  4^9 
o.o4  4o4 

9.87  062 
9.87  o5o 
9.87  039 

12 
II 

56 

55 
54 

7 
8 

9 

9.82  649 
9.82  663 
9.82  677 

14 
14 
14 

14 
14 
14 

9.95  622 
9.95  647 
9.95  672 

25 
?5 
26 

25 
25 
26 

o.o4  378 
0.04353 
0.04328 

9.87028 
9.87  016 
9.87005 

12 
II 
12 

II 
12 
II 

53 

52 

5i 

10 

9.82  691 

9.95  698 

o.o4  302 

9.86993 

60 

1 1 

12 

i3 

9.82  705 
9.82  719 
9.82  733 

9.95  723 
9.95748 
9.95774 

o.o4  277 

0.04  252 

o.o4  226 

9.86  982 
9.86  970 
9.86  959 

49 
48 

47 

i4 
i5 
i6 

9.82  747 
9.82  761 
9.82775 

14 
14 
14 

9.95799 
9.95825 
9.95  85o 

26 
25 

o.o4  201 
o.o4  175 
o.o4  150 

9,86947 
9.86936 
9.86  924 

II 
12 

46 
45 
44 

i8 
19 

9.82  788 
9.82  802 
9.82  816 

13 
14 
14 
14 
14 
14 
14 

9.95  875 
9.95  901 
9.95926 

26 

25 

26 

25 
25 
26 

25 

25 
26 

o.o4  125 
o.o4  099 
o.o4  074 

9.86913 
9.86  902 
9.86890 

II 
12 
II 

12 
12 
II 

43 
42 
4i 

40 

20 

9.82830 

g.95  952 

o.o4  o48 

9.86  879 

21 
22 
23 

9.82  844 
9.82858 
9.82  872 

9.95977 
9.96  002 
9.96  028 

0.04  023 

o.o3  998 
o.oS  972 

9.86867 
9.86  855 
9.86844 

39 
38 
37 

24 
25 

26 

9.82885 
9.82  899 
9.82  913 

14 

14 

9.96  o53 
9.96  078 
9.96  io4 

o.o3  947 
o.o3  922 
0.03896 

9.86  832 
9.86  821 
9.86  809 

II 

la 

36 
35 
34 

27 
28 
29 

9.82  927 
9.82  941 
9.82955 

J4 
14 

13 

9.96  129 
9.96  155 
9.96  180 

26 

25 
25 

o.o3  871 
0.03  845 
o.o3  820 

9.86798 
9.86786 
9.86775 

12 
II 
12 

33 

32 

3i 

30 

9.82  968 

9.96  2o5 

0.03795 

9.86  763 

30 

L.  Cos. 

d. 

L.  Cotg.  1  d. 

L.  Tang. 

L.  Sin. 

d. 

' 

47°  30 

'. 

1 

PP 

.2 
•3 

36 

as 

M 

13 

.1 
.2 
•3 

13 

II 

2.6 
7.8 

2-5 

7  5 

I 
2 
•3 

1.4 

2.8 

4-2 

1.3 

2.6 

3-9 

1-3 
2.4 
3.6 

I.I 
2.2 
33 

•4 

•5 

.6 

10.4 
13.0 

15-6 

10. 0 
12.5 
15.0 

•4 

•5 

.6 

5.6 
7.0 
8.4 

5.2 
6.5 

7.8 

•4 

•5 

.6 

4.8 
6.0 
72 

44 

u 

-.1 
■Q 

18.3 
20.8 

17- 5 

20.0 
22-5 

•7 
.8 

9.8 
11.2 
12.6 

9.1 
10.4 

•7 
.8 

8.4 
9.6 
10.8 

7-7 
8.8 

I  a 


4 

2°  30 

', 

L.  Sin. 

d. 

L.  Tang.  |  d. 

L.  Cotg.       1 

u.  Cos.     d. 

30 

9.82  968 

9.96  205 

26 
25 

o.o3  795      9 

.86  763 

II 

30 

3i 

9.82  982 

14 

9.96  23l 

o.o3  769       9 

.86  752 

29 

32 

9.82  996 

14 

9.96  256 

25 
26 

o.o3  744      9 

.86  740 

28 

33 

9.83  010 

9.96  281 

o.o3  719      9 

.86  728 

27 

34 

9.83  023 

14 

9.96  307 

25 

o.o3  693      9 

.86  717 

26 

35 

9.83  037 

14 

9.96  332 

o.o3  668      9 

.86  705 

2D 

36 

9.83  o5i 

9.96  357 

26 

o.o3  643      9 

.86694 

24 

37 

9.83  065 

13 

9.96383 

25 

o.o3  617      9 

86682 

23 

38 

9.83  07S 

9.96  4o8 

o.o3  592      9 

86  670 

22 

39 

9.83  092 

9.96433 

26 

25 
26 

o.o3  567      9 

86  659 

12 
12 

21 

20 

40 

9.83  106 

14 
13 

9.96  459 

o.o3  54i       9 

86  647 

4i 

9.83  120 

9.96484 

o.o3  5i6      9 

.86  635 

J9 

42 

9.83  i33 

9.96  5io 

0.03490      9 

.86624 

18 

43 

9.83  147 

14 

9.96  535 

25 

25 

o.o3  465      9 

.86612 

12 

17 

44 

9.83  161 

13 

9.96  56o 

26 

o.o3  44o      9 

.86  600 

16 

45 

9.83  174 

9.96  586 

o.o34i4      9 

.86  589 

i5 

46 

9.83  188 

14 

9.96  61 1 

25 
25 

0.03389      9 

.86  577 

12 

i4 

47 

9.83  202 

9.96  636 

26 

0.03  364      9 

.86  565 

i3 

48 

9.83215 

9.96  662 

0.03  338      9 

.86  554 

12 

49 

9.83  229 

9.96  687 

25 

o.o33i3      9 

.86  542 

1 1 

50 

9.83  242 

9.96  712 

25 

o.o3  288      9 

.86  53o 

10 

5i 

9.83  256 

14 
14 

9.96  738 

o.o3  262      9 

.86  5i8 

9 

52 

9.83  270 

9.96  763 

o.o3  237      9 

.86  507 

8 

53 

9.83  283 

14 

9.96  788 

25 

26 

o.o3  212      9 

.86495 

12 

7 

54 

9.83297 

13 

9.96  8i4 

o.o3i86      9 

.86  483 

6 

55 

9.83  3io 

9.96  83g 

o.o3  161       9 

.86472 

5 

56 

9.83  324 

14 
14 

9.96864 

25 
26 

o.o3  i36      9 

.86460 

12 

4 

57 

9.83  338 

9.96  890 

o.o3  no      9 

.86  448 

3 

58 

9.83  35i 

9.96915 

o.o3o85      9 

.86  436 

2 

59 

9-83  365 

14 

9.96  940 

25 
26 

o.o3o6o      9 

.86425 

I 

60 

9.83378 

13 

9.96  966 

o.o3o34      9 

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0 

L.  Cos. 

d. 

L.  Cotg.  i  d. 

L.  Tang.      1 

u.  Sin.    1  d. 

' 

47°.                                                      1 

PP 

.1 

26 

25 

14 

13 

. 

13 

II 

2.6 

2-5 

.1 

1.4 

'•? 

1.2 

I.I 

.2 

5-2 

5-0 

.2 

2.8 

2.6 

2.4 

2.2 

•3 

7.8 

7-5 

•3 

4.2 

3-9 

3.6 

3-3 

•4 

10.4 

10. 0 

■4 

5.6 

5-2 

4-8 

4.4 

■5 

13.0 

12.5 

•s 

7.0 

6.5 

^ 

6.0 

5-5 

.6 

156 

15.0 

.6 

8.4 

7-8 

.( 

7.2 

6.6 

•  7 

18.2 

'7-5 

■7 

9.8 

9.1 

8.4 

7-7 

.8 

20.8 

20.0 

8 

II. 2 

10.4 

.f 

9.6 

8.8 

22.5 

12.6 

.c 

)  1    10.8    1     9.9    1 

ii5 


43°. 

' 

L.  Sin. 

d.     ] 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.83  378 

9.96  966 

25 

25 

o.o3  o34 

9.86  4x3 

12 
17 

60 

I 

9.83  392 

13 

9.96991 

o.o3  009 

9.86  4oi 

59 

2 

9.83  4o5 

9.97  016 

0.02  984 

9.86  389 

68 

3 

9.83419 

14 

9.97  042 

25 

25 

0.02  958 

9.86  877 

^1 

4 

9.83432 

9.97067 

0.02  933 

9.86  366 

T7 

56 

6 

9.83  446' 

9.97092 

26 

0.02  908 

9.86354 

55 

6 

9.83459 

13 

9.97  118 

25 

0.02  882 

9.86  842 

54 

7 

9.83  473 

14 

9.97  i43 

0.02  857 

9.86  880 

12 

53 

8 

9.83  486 

9.97  168 

0.02  832 

9.863i8 

52 

9 

9.83  500 

14 

9.97  193 

26 

25 

25 

0.02  807 

9.86806 

II 
12 

5i 

10 

9.83  5i3 

13       ~ 

9.97219 

0.02  781 

9.86  295 

50 

1 1 

9.83  527 

'4       - 

9.97  244 

0.02  756 

9.86283 

49 

12 

9.83540 

9.97269 

26 

0.02  781 

9.86  271 

48 

i3 

9.83  554 

'3 

9.97295 

25 

0.02  705 

9.86  259 

12 

47 

i4 

9.83567 

14 

9.97  820 

25 

0.02  680 

9.86  247 

4b 

i5 

9.83  581 

9.97345 

36 

0.02  655 

9.86  235 

45 

i6 

9.83  594 

'3 

9.97371 

0.02  629 

9.86  228 

44 

'7 

9.83608 

'3 

9.97396 

25 

0.02  6o4 

9.86  211 

43 

i8 

9.83  621 

9.97  421 

26 

0.02  579 

9.86  200 

42 

19 

9.83  634 

'3 

9.97447 

25 

0.02  553 

9.86  188 

X2 

4i 

20 

9.83  648 

>4 

9.97472 

0.02  528 

9.86  176 

40 

21 

9.83  661 

13 

9.97497 

26 

0.02  5o3 

9.86  1 64 

39 

22 

9.83674 

9.97  523 

0.02  477 

9.86  l52 

38 

23 

9.83  688 

14 
13 

9.97  548 

25 

25 

0.02  452 

9.86  i4o 

12 

37 

24 

9.83  701 

9.97573 

0.02  427 

9.86  128 

86 

2b 

9.83715 

9.97598 

26 

0.02  4o2 

9.86  116 

35 

26 

9.83  728 

13 

9.97  624 

0.02  376 

9.86  io4 

34 

27 

9.83741 

9.97649 

0.02  35i 

9.86  092 

33 

28 

9.83755 

9.97674 

26 

0.02  826 

9.86  080 

82 

29 

9.83  768 

9.97700 

0.02  3oo 

9.86068 

3i 

30 

9.83  781 

13      - 

9-97725 

25 

0.02  275 

9.86  o56 

30 

L.  Cos. 

d. 

L.  Cotg. 

"dT 

L.  Tang. 

L.  Sin. 

d. 

' 

46°  3C 

y. 

1 

PP 

.1 

36 

as 

.1 

»4 

13 

.1 

13 

II 

2.6 

2-5 

1-4 

'•3 

1.3 

I.I 

.2 

5-2 

50 

.3 

2.8 

2.6 

.3 

2-4 

2.2 

•3 

7.8 

7-5 

•3 

4.2 

3-9 

•3 

3-6 

3-3 

•4 

10.4 

lao 

•4 

5.6 

S» 

•4 

4.8 

4-4 

•  5 

13.0 

ia.5 

•s 

7.0 

6.5 

•5 

6.0 

5  5 

.6 

.5.6 

iS-o 

.6 

8.4 

7.8 

.6 

7.3 

6.6 

.7 

18.2 

'75 

•7 

9.8 

9.1 

•7 

8.4 

7.7 

.8 

20.8 

20.0 

.8 

II. 2 

10.4 

.8 

9.6 

8.8 

.2.6 

^^ 

.9          10.8 

116 


43 

°30 

', 

' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

g.83  781 

14 
13 

9-97  T^b 

25 
26 

0.02  275 

9.86  o56 

12 

30 

3 1 

y.83  795 

9.97  75o 

0.02  250 

9.86044 

29 

32 

9.83  808 

9.97776 

0.02  224 

9.86  o32 

28 

33 

9.83  821 

13 

9.97  801 

25 

0.02  199 

9.86  020 

27 

34 

9.83  834 

9.97  826 

0.02  174 

9.86  008 

26 

35 

9.83  848 

9.97851 

26 

0.02  149 

9.85  996 

25 

3(j 

9.83  861 

13 

9.97877 

0.02   123 

9.85  984 

24 

^57 

9.83  874 

9.97902 

0.02  098 

9.85  972 

23 

38 

9.83  887 

9.97  927 

26 
25 

25 
26 

0.02  073 

9.85  960 

22 

39 

9.83  901 

14 
13 

13 

9.97  953 

0.02  047 

9.85948 

12 
12 

21 

40 

9.83  914 

9.97  978 

0.02  022 

9.85  936 

20 

4i 

9.83  927 

9.98  oo3 

o.oi  997 

9-85  924 

J9 

42 

9.83  940 

9.98  029 

o.oi  971 

9.85  912 

18 

43 

9.83954 

14 

g.98  o54 

0.01  946 

9.85  900 

•7 

44 

9.83  967 

9.98079 

25 
26 

O.OI  921 

9.85  888 

16 

45 

9.83  980 

9.98  io4 

0.01  896 

9.85  876 

i5 

46 

9.83  993 

'3 

9.98  i3o 

O.OI  870 

9.85  864 

i4 

«3 

25 

M 

4? 

9.84  006 

9.98  i55 

25 

O.OI  845 

9.85  85i 

i3 

48 

9.84  020 

9.98  180 

O.OI  820 

9.85  839 

12 

49 

9.84  o33 

'3 
13 

13 
13 

9.98  206 

25 

25 

O.OI  794 

9.85  827 

12 
12 

I  I 
10 

9 

50 

9.84  o46 

9.98  23l 

O.OI  769 

9.85  8i5 

5i 

9,84  059 

9.98  256 

O.OI  744 

9.85  8o3 

52 

9.84  072 

9.98  281 

26 

O.OI  719 

9.85  791 

8 

53 

9.84  o85 

13 

9.98  307 

O.OI  693 

9.85  779 

7 

•3 

25 

13 

54 

9.84098 

9.98  332 

25 

O.OI  668 

9.85  766 

'    6 

55 

9.84  112 

9.98  357 

O.OI  643 

9.85754 

5 

56 

9.84  125 

13 
13 

9.98  383 

25 

O.OI  617 

9.85  742 

12 

4 

5? 

9.84  1 38 

9.98  4o8 

O.OI  592 

9.85  730 

3 

58 

9.84  i5i 

9.98433 

O.OI  567 

9.85  718 

2 

59 

9.84  1 64 

•3 
13 

9.98  458 

26 

O.OI  542 

9.85  706 

13 

I 

60 

9.84  177 

9.98484 

O.OI  5i6 

9.85  693 

0 

L.  Cos. 

d. 

L.  Cotg.      d. 

L.  Tangr. 

L.  Sin. 

d.' 

46°.                                                       1 

PP 

.1 

26 

35 

.1 

14 

13 

.1 

13 

2.6 

2-5 

1.4 

1-3 

1.2 

.2 

5-2 

50 

.2 

2.8 

2.6 

.2 

2.4 

■3 

7-8 

75 

•3 

4-2 

3-9 

•3 

3.6 

•4 

10.4 

10.0 

•4 

5-6 

5.2 

•4 

4.8 

•5 

13.0 

12.5 

■  s 

7.0 

6-5 

•5 

6.0 

.6 

•5.6 

15.0 

.6 

8.4 

7.8 

.6 

7.2 

•7 

18.2 

17-5 

•7 

Q.8 

9.1 

■7 

8.4 

.8 

20.8 

20.0 

.8 

I 

1.2 

10.^ 

.8 

9.6 

2.6             1              IT. 7 

^_ 

10.8 

117 


44°. 


/ 

L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.84  177 

'3 
'3 
«3 

9.98484 

25 
25 
26 
25 
25 
25 
26 

o.oi  5i6 

9.85693 

12 
12 

60 

I 

2 

3 

g.84  190 
9.84203 
9.84  216 

9.98  509 
9.98  534 
9.98  56o 

o.oi  491 
0.01  466 
o.oi  44o 

9.85  681 
9.85  669 
9.85  657 

59 
58 
57 

4 
5 
6 

9.84  229 
9.84  242 
9.84255 

J3 
•3 
14 
13 
'3 
13 
'3 
•3 
13 

9.98585 
9.98  610 
9.98635 

O.OI  4i5 
O.OI  390 
O.OI  365 

9-85  645 
9.85  632 
9.85  620 

«3 
12 

56 
55 
54 

7 

8 

9 

9.84  269 
9.84  282 
9.84  295 

9.98  661 
9.98  686 
9.98  711 

25 
25 
26 

25 
25 
25 
26 

25 
25 

O.OI  339 
O.OI  3i4 
0.01  289 

9.85  608 
9.85596 
9.85  583 

12 

'3 
12 

12 
12 

»3 

53 

52 

5i 

10 

9.84  3o8 

9.98  737 

O.OI  2G3 

9.85571 

50 

4y 
4s 
47 

1 1 

12 

i3 

9.84  321 
9.84334 
9.84347 

9.98  762 
9.98787 
9.98  812 

O.OI  238 

O.OI    2l3 

O.OI   188 

9.85  559 
9.85  547 
9.85534 

i4 
i5 
i6 

9.84360 
9.84373 
9.84  385 

13 
12 

9.98  838 
9.98863 
9.98888 

O.OI  162 
O.OI  i37 
O.OI  112 

9.85  522 

9.85  5io 
9.85  497 

12 

13 

46 

45 

44 

«7 
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'9 

9.84  398 
9.84411 
9.84  424 

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13 

'3 
13 
«3 

9.98  913 
9.98939 
9.98  964 

26 
25 
25 
26 

25 

25 

O.OI  087 
O.OI  061 
0.01  o36 

9.85485 
9.85473 
9.85  46o 

12 

«3 

12 
12 

13 
12 

43 
42 
4i 

20 

9.84437 

9.98989 

O.OI  01 1 

9.85  448 

40 

■2  1 
2  2 
23 

9.84  45o 
9.84  463 
9.84476 

9.99015 
9.99  o4o 
9.99  o65 

0.00  985 
0.00  960 
0.00  935 

9.85  436 
9.85423 
9.85  4ii 

39 
38 
37 

24 
25 

26 

27 
28 
29 

9.84489 
9.84  5o2 

9.84515 

9.84528 
9.84540 
9.84553 

13 
13 
13 
12 

'3 
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9.99090 
9 . 99  116 
9.99  i4i 

9.99  166 
9.99  191 
9.99217 

26 
25 

25 

25 

36 

25 

0 .  00  9 1 0 
0.00  884 
0.00  859 

0.00834 
0.00  809 
0.00  783 

9.85  399 
9.85  386 
9.85374 

9.85  36i 
9.85  349 
9.85  337 

13 
12 

13 
12 
12 

'3 

36 
35 
34 

33 

32 

3r 
30 

30 

9.84  566 

9.99  242 

0.00  758 

9.85  324 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

45°  30 . 

P 

P 

I 
1 
3 

36 

»5 

.1 
.2 

.3 

14 

13 

.1 
.2 

•3 

12 

2.6 

5-2 

7.8 

2-5 

so 
7-S 

■•4 
1.8 
♦•2 

1:1 

3-9 

1.2 
2.4 
3.6 

4 
5 

6 

10.4 
13.0 
15.6 

lO.O 

12.5 
15.0 

•4 
•s 

.6 

5.6 
8.4 

5.2 
6.5 
7.8 

•4 

•5 

.6 

4.8 
6.0 

7-2 

I 

18.3 
20.8 

17-5 
20.0 
22.5 

■7 
.8 

I 
> 

5.8 
1.2 

2.6 

9' 
10.4 

•7 

.8 

10.8 

118 


44°  30'. 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

25 

26 

25 

25 

L.  Cotg. 

L.  Cos. 

d. 

"so' 

9.84  566 

13 
13 
13 

J3 
12 

13 

9 

99  242 

0.00  768 

9.S5  324 

12 

13 
12 

30 

3i 

32 

33 

9.84579 
9.84  592 
9.84605 

9 
9 
9- 

99267 
99293 
99  3i8 

0.00  733 
0.00  707 
0.00  682 

9.85  312 
9.85  299 
9.85  287 

29 
28 
27 

34 
35 
36 

9.84618 
9.84  63o 
9.84643 

9 
9 
9- 

99343 
99  368 
99  394 

25 
26 

0.00  657 
0.00  632 
0.00  606 

9.85  274 
9-85  262 
9.85  250 

12 
12 

26 

25 

24 

37 
38 

39 

9.84  656 
9.84  669 
9.84  682 

'3 
13 
12 

13 
13 
13 
12 

13 
13 

9 

9- 

9- 

99419 
99  444 
99  469 

25 

25 

26 

25 

25 

25 

26 
25 
25 

0.00  58 1 
0.00  556 
0.00  53i 

9-85  237 
9.85  225 
9.85  212 

'3 
12 

13 
12 

13 
12 
13 
12 

13 
12 

23 
22 
21 

40 

9.84694 

9- 

99  495 

0.00  5o5 

9.85  200 

20 

4i 

42 

43 

44 
45 
46 

9.84  707 
9.84  720 
9.84733 

9.B4745 
9.84758 
9.84  771 

9- 
9- 
9- 

9- 
9- 

9- 

99  520 
99  545 
99570 

99  596 
99  621 
99  646 

0.00  480 
0.00  455 
0.00  43o 

0.00  4o4 
0.00  379 
0.00  354 

9.85  187 
9.85  175 
9.85  162 

9.85  150 
9.85  137 
9.85  125 

19 

18 

17 
16 
i5 

i4 

47 
48 

49 

9.84784 
9.84  796 
9.84  809 

12 
13 
13 

13 
12 

13 
13 
12 

13 

9- 
9- 
9- 

99  672 
99697 
99  722 

25 
25 
25 
26 
25 
25 
25 
26 

25 

0.00  328 
0.00  3o3 
0.00  278 

9.85  112 
9.85  100 
9-85  087 

'3 
12 
13 
'3 
12 
13 
12 
13 
12 

'3 

i3 
12 
1 1 

50 

9.84822 

9- 

99747 

0.00  253 

9.85  074 

10 

5i 

52 

53 

54 
55 
56 

9.84835 

9-84  847 
9.84860 

9.84  873 
9.84  885 
9.84898 

9- 
9- 
9- 

9- 
9- 
9- 

99773 
99798 
99  823 

99848 
99874 
99899 

0.00  227 
0.00  202 
0.00  177 

0.00  l52 
0.00  126 
0.00  1 01 

9-85  062 
9.85  049 
9.85  037 

9.85  024 
9.85  012 
9.84999 

9 

8 

7 
6 
5 
4 

57 
58 
59 

9.84  911 
9.84  923 
9.84  936 

12 
13 

13 

9- 
9- 
9- 

99  924 
99949 
99  975 

25 

26 
25 

0.00  076 
0.00  o5i 
0.00  025 

9.84  986 
9.84  974 
9.84  961 

12 
13 
12 

3 

2 

1 

60 

9.84  949 

0. 

00  000 

0.00  000 

9.84  949 

0 

L.  Cos. 

d. 

L.  Cotg.  1 

d. 

L.  Tang. 

L.  Sin. 

d. 

/ 

4 

-.5°. 

PP 

.1 

.2 

•3 

36 

25 

14 

13 

.1 
.2 

•3 

13 

2.6 
5.2 
7.8 

2 
5 
7 

s 
0 

5 

.1 

.2 
•3 

, 

•4 

.8 

(.2 

1-3 
2.6 
3-9 

1.2 
2.4 

3-6 

•4 

■5 
.6 

10.4 
13.0 

.5-6 

10 
12 
15 

0 

5 
0 

•4 
.5 
.6 

I 

.6 
.0 
•4 

5-2 

6.5 
7.8 

•4 
•5 

.6 

4.8 
6.0 
7-2 

•7 
.8 

18.2 
208 

17 
20 

22 

s 

0 

■7 

.8 

c 
11 

12 

.8 
.2 
.6 

9.1 
10.4 

•7 

.8 

8.4 
9.6 
10.8 

119 


TABLE   III 

FIVE-PLACE    LOGARITHMS 

OF  THE 

SINE    AND    TANGENT    OF 
SMALL    ANGLES 

THE  SINE  AND  TANGENT  TO  EVERY  SECOND  FROM  0°  TO  8'  ;  TO  EVERY 
TEN   SECONDS   FROM   0°   TO   2°. 

THE  COSINE  AND  COTANGENT  TO  EVERY  SECOND  FROM  90°  TO  89"* 
52';  TO  EVERY  TEN   SECONDS  FROM  90°  TO   88°. 


FUNCTIONS   OF  SMALL  ANGLES. 


0°. 

LOGARITHMIC 

SINK 

AND 

TANGENT 

'" 

0" 

1" 

2" 

3" 

4" 

5" 

6" 

7" 

8" 

9" 

10" 

Oo 

lO 

20 

5-68557 
98660 

68557 
72697 

*oo779 

98660 

76476 

4,02800 

^16270 

799b  2 
*0473« 

,28763 
83170 
»o6579 

•38454 

86167 

•0835. 

•46373 
88969 

«'oo55 

»53o67 
91602 
,11694 

,58866',63982 

94085 1  96433 

•'3273  *'4797 

»6B557 
9S660 
,1627c 

50 
40 
30 

30 
40 

5° 

6.  16270 
28763 
38454 

17694 
29836 
39315 

19072 
30882 
40158 

20409 
3 '904 
40985 

21705 
32903 
4 '797 

22964 
33879 
42594 

24188 
34833 
43376 

25378 
35767 
44'45 

26536 
36682 
44900 

27664 

37577 
45643 

28763 
38454 
46373 

20 
10 
0  59 

1  0 

10 
20 

6.46373 

6. 5  3067 

8866 

7090 
3683 
9406 

7797 
4291 

9939 

8492 

4890 

♦0465 

9'75 

5481 

♦0985 

9849 
6064 

*M99 

*o5>2 

6639 

4,2007 

♦  "65 
7207 

•2509 

4,1808 

7767 

4,3006 

*2442 

832c 
♦3496 

4,3067 

8866 

*3982 

50 
40 
30 

30 
40 

50 

6.63982 

8557 

6.72697 

4462 
8990 
3090 

4936 
9418 

3479 

5406 
9841 
3865 

5870 

^026 1 

4248 

6330 

4,0676 

4627 

6785 

4,1088 

5003 

7235 

•  '496 

5376 

7680 

4,1900 

5746 

8l21 

4,230c 

6112 

8557 

4,2697 

6476 

20 
10 
o58 

2  0 

10 
20 

6476 

9952 

6.83170 

6836 

,0285 

3479 

7>93 

*o6i5 

3786 

7548 

*0943 

4091 

7900 

*i268 

4394 

8248 

*i59' 
4694 

8595 
4,1911 

4993 

8938 

4,2230 

5289 

9278 

*2545 

5584 

9616 

•2859 

5876 

9952 

♦317c 

6167 

50 
40 
30 

30 
40 

50 

6.67 

8969 

6.9  1602 

6455 
9240 

•857 

6742 
9509 
2110 

7027 
9776 
2362 

7310 

#0042 

2612 

759' 

4,0306 

2861 

7870 

4.0568 

3109 

8.47 

4,0829 

3355 

8423 

4,1088 

3599 

8697 

*'346 
3843 

8969 

4,1602 

4085 

20 
10 
0  57 

3  0 

10 
20 

4085 

6433 
8660 

4325 
6661 

8877 

4565 
6888 
9093 

4803 
7113 
9307 

5039 
7338 
9520 

5275 
7561 

9733 

5509 
7783 
9944 

5742 

8004 

•0155 

5973 
8224 

•0364 

6204 

8443 

♦0572 

6433 

8660 

#0779 

50 
40 
30 

30 
40 

50 

7.00779 
2800 
4730 

0986 
2997 
4919 

1191 
3 '93 
5106 

1395 
3388 
5293 

1599 
3582 

5479 

1801 
3776 
5664 

2003 
3968 
5849 

2203 
4160 
6032 

2403 
435' 
6215 

2602 

4541 
6397 

280a 
4730 
6579 

20 
10 
0  66 

4  0 
10 

20 

6579 

835' 

7- 1 0055 

6759 
8525 
0222 

6939 
8698 
0388 

7118 

887c 
0553 

7296 
9041 
0718 

7474 
9211 

0882 

765' 
9381 
1046 

7827 
955' 
1209 

8003 
9719 
137' 

8177 
9887 
•533 

835' 

*oo55 

1694 

50 
40 
30 

30 
40 
5° 

1694 
3273 
4797 

1854 
3428 
4947 

2014 
3582 
5096 

2174 

3736 
5244 

2333 
3889 
5392 

2491 
4042 
5540 

*  2648 
4194 
5687 

2805 
4346 
5833 

2962 
4497 
5979 

3118 

4647 
6125 

3273 
4797 
6270 

20 
10 
055 

5  0 

10 

20 

7. 1  6270 
7694 
9072 

6414 

7834 
9208 

6558 
7973 
9343 

6702 
8112 
9478 

6845 
8250 
9612 

6987 
8389 
9746 

7130 
8526 
9879 

7271 

8663 

4,0012 

7413 

8800 

4,0145 

7553 

8937 

*o277 

7694 

9072 

4,0409 

50 
40 
30 

30 
40 

5° 

7. 2  0409 
1705 
2964 

0540 
•833 
3088 

0671 
i960 
3212 

0802 
2087 
3335 

0932 
2213 
3458 

1062 
2339 
3580 

1191 
2465 
3702 

1320 
2590 
3824 

1449 
2715 
3946 

'577 
284c 
4067 

1705 
2964 
4188 

20 
10 
054 

6  0 

10 
20 

4188 
5378 
6536 

4308 
5495 
6650 

4428 
5612 
6764 

4548 
5728 
6877 

4668 

584s 
6991 

4787 
596. 
7104 

4906 
6076 
7216 

5024 
6192 
7329 

5'42 
6307 

744' 

5260 
6421 
7552 

5378 
6536 
7664 

50 
40 

30 

30 
40 

5" 

7664 

8763 
9836 

7775 
8872 
9942 

7886 

8980 

#0047 

*o>52 

8107 

9196 

•0257 

8217 

9303 

410362 

8327 

9410 

♦0467 

8437 

95'7 

*o57' 

8546 

9623 

•0675 

8655 
973c. 

*0779 

8763 

9836 

4,0882 

20 
10 
0  53 

7  0 

10 
20 

7.30882 
1904 
2903 

0986 
2005 
3001 

1089 
2106 
3100 

II9I 
2206 
3'98 

1294 
2306 
3296 

1396 
2406 
3393 

1498 
2506 

349' 

1600 
2606 
3588 

1702 
2705 
3685 

1803 
2804 
3782 

1904 
2903 
3879 

50 
40 
30 

30 
40 

3879 
483;^ 
576' 

3975 
5860 

4071 
5022 
5952 

4167 
5.16 
6044 

4263 
5209 
6135 

4359 
5303 
6227 

4454 
539^ 
63.8 

4549 
5489 
6409 

4644 
5582 
6500 

4739 
5675 
6591 

4833 

5767 
6682 

20 
10 
0  52 

10" 

9" 

8" 

7" 

6" 

5" 

4" 

3" 

2" 

1  " 

0" 

" 

LOGARITHMIC  COSINE  AND  COTANGENT. 


89= 


FUNCTIONS    OF    SMALL    ANGLES. 
0°. 


L.  Sin.    L.  Tang.f           | 

,   „ 

L.  Sin.   L.  Tang. 

0  o 

10 
20 

3o 
4o 
5o 

5.68  557 
5.98660 

6. 16  270 
6.28763 
6.38  454 

5.68  557 
5.98660 
6. 16  270 
6.28763 
6.38  454 

0  60 

5o 
4o 
3o 
20 
10 

7  3o 
4o 
5o 

7.33879 

7.34  833 

7.35  767 

7.33879 
7.34  833 
7.35767 

3o 
20 
10 

8  0 

10 

20 

3o 
4o 
5o 

7.36  682 
7.37577 
7-38  454 

7.39314 

7.40  i58 
7.40  985 

7.36  6»2 
7.37577 
7.38455 
7-39  315 
7.40  i58 
7.40  985 

0  52 

5o 
4o 
3o 
20 
10 

1     0 

10 
20 

3o 

4o 
5o 

6.46  373 
6.53  067 
6.58  866 
6.63  982 
6.68  557 
6.72  697 

6.46373 
6.53  067 
6.58  866 

6.63982 
6.68  557 
6.72  697 

0  59 

5o 
4o 
3o 
20 
10 

9  0 

10 
20 

3o 
4o 
5o 

7.41  797 

7.42  594 
7.43376 

7.44  145 
7.44900 

7.45  643 

7.41  797 
7-42  594 
7.43376 

7.44  i45 

7.44  900 

7.45  643 

0  51 

5o 
4o 
3o 
20 
10 

2  o 

ID 

20 

3o 

4o 
5o 

6.76476 
6  79  952 
6.83  170 

6.86  167 
6.88969 
6.91  602 

6.76476 
6.79  952 
6.83  170 

6.86  167 
6.88969 
6.91  602 

0  58 

5o 
4o 

3o 
20 
10 

10  0 

10 
20 
3o 
4o 
5o 

7.46373 
7.47  090 
7-47  797 
7-48  491 
7.49  175 
7-49  849 

7.46373 
7-47  091 
7-47  797 
7.48492 
7-49  176 
7-49  849 

0  50 

5o 
4o 
3o 
20 
10 

3  o 

10 
20 

So 
do 
5o 

6.94  085 
6.96  433 
6.98  660 

7.00779 
7.02  800 
7.04  73o 

6.94  085 
6.96433 
6.98  660 

7.00779 
7.02  800 
7.04  730 

0  57 

5o 
4o 
3o 
20 
10 

11  0 

10 
20 

3o 
4o 
5o 

12  0 

10 
20 

3o 
4o 
5o 

7.5o  5i2 
7-5i  165 
7.5i  808 

7.52  442 

7.53  067 
7.53  683 

7.50  5i2 
7.5i  i65 

7. 5 1  809 

7.52443 
7.53  067 
7.53  683 

0  49 

5o 
4o 
3o 
20 
10 

4  0 

lO 
20 

3o 
4o 
5o 

7.06  579 
7.08  35i 

7.10  055 

7. 1 1  694 
7.13  273 
7.14797 

7.06  579 
7.08  352 

7.10  055 

7. 11  694 
7.13273 
7.14  797 

0  56 

5o 
4o 

3o 
20 
10 

7.54  291 
7.54  890 
7.55481 

7.56  064 

7.56  639 

7.57  206 

7.54  291 
7.54  890 
7-55  481 

7-56  064 

7.56  639 

7.57  207 

0  48 

5o 
4o 

3o 
20 
10 

5  o 

lO 
20 

3o 
4o 
5o 

7.16  270 

7.17  694 

7. 19  072 

7.20  409 

7.21  7o5 

7.22  964 

7. 16  270 

7.17  694 
7.19073 

7.20  409 

7.21  70' 

7.22  964 

0  55 

5o 
4o 
3o 
20 
10 

13  0 

10 
20 

3o 
4o 
5o 

7.57  767 

7.58  320 
7-58  866 

7.59  4o6 
7.59939 

7.60  465 

7.57767 

7.58  320 
7.58867 

7.59  4o6 
7.59939 

7.60  466 

0  47 

5o 
4o 
3o 
20 
10 

6  o 

lO 
20 

3o 

4o 
5o 

7.24  188 

7.25  378 

7.26  536 

7.27  664 

7.28  763 

7.29  836 

7.24  188 

7.25  378 

7.26  536 

7. 87  664 

7.28  764 

7.29  836 

0  54 

5o 
4o 
3o 
20 
10 

14  0 

10 
20 

3o 
4o 
5o 

7.60  985 

7.61  499 

7.62  007 

7.62  Sog 

7.63  006 
7.63  496 

7-60  986 

7.61  500 

7.62  008 
7.62  5io 
7-63  006 
7-63  497 

0  46 

5o 
4o 

3o 
20 
10 

7  o 

ID 

20 

3o 

7.30  882 
7.  3 1  904 

7.32  903 

7.33  879 

7.30882 

7. 3 1  904 

7.32  903 
7.33879 

0  53 

5o 
4o 

3o52 

15  0 

7.63  982 

7-63  982 

0  45 

L.  Cos. 

L.  Cotg. 

.,    , 

L.  Cos. 

L.  Cotg. 

■'     ' 

123 


89°. 


FUNCTIONS    OF    SMALL    ANGLES. 


1      It 

L.  Sin. 

L.  Tang. 

15  o 

7.63  9»2 

7.63  982 

0  45 

lO 

7.64461 

7.64  462 

5o 

20 

7.64936 

7.64937 

4o 

3o 

7.65406 

7.65  4o6 

3o 

4o 

7.65  870 

7.65871 

20 

5o 

7.66  33o 

7.66  33o 

lO 

16  0 

7.66784 

7.66785 

0  44 

lo 

7.67235 

7.67  235 

5o 

20 

7.67680 

7.67  680 

4o 

3o 

7.68  121 

7.68  121 

3o 

4o 

7.68557 

7.68  558 

20 

5o 

7.68989 

7.68  990 

10 

17  0 

7.69  417 

7.69  4io 

0  43 

10 

7.69841 

7.69  842 

5o 

20 

7.70  261 

7.70  261 

4o 

3o 

7.70  676 

7.70677 

3o 

4o 

7.71  088 

7.71  088 

20 

5o 

7.71  496 

7.71  496 

10 

18  0 

7.71  900 

7.71  900 

0  42 

10 

7.72  3oo 

7.72  3oi 

5o 

20 

7.72697 

7.72697 

4o 

3o 

7.73  090 

7.73  090 

3o 

4o 

7.73479 

7.73  480 

20 

5o 

7.73865 

7.73866 

10 

19  0 

7-74  248 

7.74248 

0  41 

10 

7.74  627 

7.74  628 

5o 

20 

7.75  oo3 

7.75  oo4 

4o 

3o 

7.75376 

7.75377 

3o 

4o 

7.75745 

7.75  746 

29 

5o 

7.76  112 

7.76  n3 

10 

20  0 

7.76  475 

7-76  476 

0  40 

10 

7.76836 

7.76837 

5o 

20 

7.77  193 

7.77  194 

4o 

3o 

7-77  548 

7.77549 

3o 

4o 

7.77899 

7.77900 

20 

5o 

7.78248 

7.78  249 

10 

21  0 

7-75*594 

7.78595 

0  39 

10 

7.78938 

7.78938 

5o 

20 

7.79278 

7.79279 

4o 

3o 

7.79  616 

7.79617 

3o 

4o 

7.79952 

7.79952 

20 

5o 

7.80284 

7.80  285 

ID 

22  0 

7.80615 

7.80615 

0  38 

10 

7.80  942 

7.80943 

5o 

20 

7.81  268 

7.81  269 

4o 

3o 

7.81  591 

7.8t  591 

3o37 

L;  Cos.  '  L.  Cotg.  1 

"     ' 

,     „ 

L.  Sin. 

L.  Tang. 

22  3o 

7.81  591 

7.81  691 

3o 

4o 

7.81  911 

7.81  912 

20 

5o 

7.82  229 

7.82  2  3o 

10 

23  0 

7.82545 

7.82546 

0  37 

10 

7.82859 

7.82  860 

5o 

20 

7.83  170 

7.83  171 

4o 

3o 

7-83479 

7.83480 

3o 

4o 

7.83786 

7.83787 

20 

5o 

7.84  091. 

7.84  092 

10 

24  0 

7.84  393 

7-84394 

0  36 

10 

7-84694 

7.84695 

5o 

20 

7.84  992 

7-84  993 

4o 

3o 

7.85  289 

7.85  290 

3o 

4o 

7.85  583 

7.85  584 

20 

5o 

7.85876 

7-85  877 

10 

25  0 

7.86  166 

7.86  167 

0  35 

10 

7-86455 

7.86  456 

5o 

20 

7  86741 

7.86743 

4o 

3o 

7.87  026 

7.87  027 

3o 

4o 

7.87  309 

7.87310 

20 

5o 

7 .87  590 

7.87591 

10 

26  0 

7.87  870 

7.87871 

0  34 

10 

7.88  147 

7.88  i48 

5o 

20 

7.88423 

7.88424 

4o 

3o 

7.88697 

7.88698 

3o 

4o 

7.88969 

7.88970 

20 

5o 

7.89  240 

7.89  24 1 

10 

27  0 

7.89  509 

7.89  5io 

0  33 

10 

7.89776 

7.89777 

5o 

20 

7.90  o4 1 

7.90  043 

4o 

3o 

7.90  3o5 

7.90  307 

3o 

4o 

7.90  568 

7.90  569 

20 

5o 

7.90  829 

7.90  83o 

10 

28  0 

7.91  088 

7.91  089 

0  32 

10 

7.91  346 

7.91  347 

5o 

20 

7.91  602 

7.91  6o3 

4o 

3o 

7.91  857 

7.91  858 

3o 

4o 

7.92  1 10 

7.92  III 

20 

5o 

7.92  362 

7.92  363 

10 

29  0 

7.92  612 

7.92613 

0  31 

10 

7.92  861 

7.92  862 

5o 

20 

7.93  108 

7.93  1 10 

4o 

3o 

7.93354 

7.93  356 

3o 

4o 

7.93599 

7.93  601 

20 

5o 

7.93  842 

7-93  844 

10 

30  0 

7-94  o84 

7.94  086 

0  30 

L.  Cos. 

L.  Cotg. 

"    ' 

124 


89°. 


FUNCTIONS    OF    SMALL    ANGLES. 
0°. 


,    „ 

L.  Sin. 

L.  Tang. 

,     „ 

L.  Sin. 

L.  Tang. 

30  0 

10 
20 

3o 

4o 
5o 

7.94  084 
7.94325 
7.94  564 

7.94  802 

7.95  039 
7.95  274 

7 
7 
7 
7 

7 

7 

94086 
94  326 

94  566 

94804 

95  o4o 
95  276 

0  30 

5o 
4o 
3o 
20 
10 

37  3o 
4o 
5o 

8.o3  775 

8.03  967 

8.04  i59 

8  .o3  777 

8.03  970 

8.04  162 

3o 
20 
10 

38  0 

10 
20 
3o 

4o 
5o 

8.o4  35o 
8.04540 

8.04  729 
8.04918 

8.05  io5 
8.o5  292 

8.04353 
8.04543 
8.o4  732 

8.04  921 

8.05  108 
8.o5  295 

0  22 

5o 
4o 
3o 
20 
10 

31  o 

10 
20 

3o 
4o 
5o 

7.95  5o8 

7-95  74i 
7.95^3 

7.96  2o3 

7.96  432 
7.96  660 

7 
7 
7 

7 
7 
7 

96  5io 
95743 
95974 
96  205 
96434 
96  662 

0  29 

5o 
4o 
3o 
20 
10 

39  0 

10 
20 

3o 

4o 
5o 

8.05478 
8.05  663 

8.05  848 

8.06  o3i 
8.06  2l4 
8.06396 

8.o5  48i 
8.05  666 

8.05  85i 

8.06034 

8.06  217 
3 .06  399 

0  21 

5o 
4o 
3o 
20 
10 

32  o 

ID 
20 

3o 
4o 
5o 

7.96  887 

7.97  ii3 
7.97  337 

7.97  56o 

7.97  782 

7.98  oo3 

7 
7 
7 

7 
7 
7 

96  889 

97  1 14 
97339 

97  562 
97784 

98  oo5 

0  28 

5o 
4o 

3o 
20 
10 

40  0 

10 
20 
3o 
4o 
5o 

8.06578 

8.06  758 
8.06938 

8.07  117 
8.07  295 
8.07473 

8.06  58i 
8.06  761 

8.06  941 

8.07  120 
8.07  298 
8.07  476 

0  20 

5o 
4o 
3o 
20 
10 

33  o 

10 
20 

3o 
4o 
5o 

7.98  223 

7.98  442 

7.98  660 

7.98876 
7.99092 

7.99  3o6 

7 
7 
7 

7 

7 
7 

98  225 
98444 
98662 

98878 

99  094 
99  3o8 

0  27 

5o 
4o 
3o 
20 
10 

41  0 

10 
20 

3o 
4o 
5o 

8.07  650 
8.07826 

8.08  002 
8.08  176 
8.08  35o 
8.08  524 

8.07653 

8.07  829 

8.08  005 
8.08  180 
8.08354 
8.08  527 

0  19 

5o 
4o 
3o 
20 
10 

34  o 

ID 
20 

3o 
4o 
5o 

7.99  620 
7.99  732 
7.99943 

8.00  1 54 
8.00  363 
8.00  571 

7 
7 
7 

8 
8 
8. 

99  522 

99734 
99  946 
00  1 56 
00  365 
00  574 

0  26 

5o 
4o 
3o 
20 
10 

42  0 

10 
20 

3o 
4o 
5o 

8.08696 
8.08868 
8 .  09  o4o 
8 .  09  2 1 0 
8.09  38o 
8.09  550 

8.08  700 

8.08  872 
8.09043 

8.09  2 14 
8.09  384 
8.09553 

0  18 

5o 
4o 
3o 
20 
10 

35  o 

10 
20 

3o 
4o 
5o 

8  .00  779 

8.00  985 

8.01  190 

8.01  395 
8.01  598 
8.01  801 

8. 
8. 
8. 

8. 
8. 
8 

00  781 

00  987 

01  193 

01  397 
01  600 
01  8o3 

0  25 

5o 
4o 
3o 
20 
10 

43  0 

ID 
20 

3o 
4o 
5o 

44  0 

10 
20 

3o 
4o 
5o 

8.09  718 
8.09886 
8.10054 

8.10  220 
8.I0386 
8.I0552 

8.09  722 

8.09  890 

8. 10  067 

8. 10  224 
8. 10  390 
8.10555 

0  17 

5o 
4o 
3o 
20 
10 

36  o 

ID 
20 

3o 

4o 
5o 

8.02  002 
8.02  203 

8.02  4o2 

8.02  601 
8.02  799 
8.02  996 

8 
8 
8 
8 
8 
8 

02  oo4 
02  2o5 
02  405 

02  6o4 
02  801 
02  998 

0  24 

5o 
4o 
3o 
20 
10 

8.10717 
8.10881 
8. 1 1  o44 
8.  II  207 
8. II  370 
8. II  53i 

8. 10  720 
8.10884 

8.11  o48 

8.11  211 
8. II  373 
8. II  535 

0  16 

5o 
4o 
3o 
20 
10 

37  o 

lO 
20 

3o 

8.o3  192 
8.o3  387 
8,o3  58i 
8.o3  775 

8 
8 
8 
8 

o3  194 
o3  390 

o3  584 

03777 

0  23 

5o 
4o 

3o22 

45  0 

8.  II  693 

8.11  696 

0  15 

L.  Cos. 

L 

.  Cotg. 

"    ' 

L.  Cos. 

L.  Cotg. 

"     ' 

125 


89°. 


FUNCTIONS    OF    SMALL    ANGLES. 

o°. 


L.  Sin. 

L.Tang. 

45  o 

lo 

20 

3o 
4o 
5o 

«.  11  6y3 

8.11  853 

8. 12  oi3 
8.12  172 
8.12331 
8.12  489 

8.  I  I  6c;6 

8.11  85? 

8. 12  017 
8.12  176 
8.12335 
8.12493 

0  15 

5o 
4o 
3o 
20 
10 

48  o 

10 
20 

3o 
4o 
5o 

8.12647 
8.12  8o4 
8.12  961 

8.i3  117 
8.i3  272 
8.13427 

8. 12  65i 
8.12808 
8.12965 
8.i3  121 
8.13276 
8.i343i 

0  14 

5o 
4o 
3o 
20 
10 

47  0 

lO 
20 

3o 
4o 
5o 

8.i3  58i 
8.13735 
8.13888 

8.i4o4i 
8,i4  193 
8.14344 

8.13585 
8.13739 
8.13892 

8.14045 
8.i4i97 

8.14  348 

0  13 

5o 
4o 
3o 
20 
10 

48  o 

10 
20 

3o 
4o 
5o 

8.14495 
8.14  646 
8.i4  796 
8.14945 
8.15094 
8.i5  243 

8.  i4  500 
8.i4  65o 
8.i4  8oo 

8.14950 
8.  i5  099 
8.15247 

0  12 

5o 
4o 
3o 
20 
10 

49  o 

10 
20 

3o 
4o 
5o 

8.15391 
8.15538 
8.15685 
8.15832 
8.15978 

8.16   123 

8.i5  395 
8.15543 
8.15690 

8. 15836 
8,15982 
8.16  128 

0  11 

5o 
4o 
3o 
20 
10 

50  o 

lO 
20 

3o 
4o 
5o 

51  o 

lO 
20 

3o 
4o 
5o 

8.16268 
8.i64i3 
8.16557 
8.16  700 
8.16843 
8.16986 

8.16273 
8.16417 
8.i656i 

8.16705 
8.16  848 
8. 16  991 

0  10 

5o 
4o 
3o 
20 
10 

S.17  128 
8.17  270 
8.17411 
8.17552 
8.17  692 
8.17832 

8.17  i33 
8.17275 
8.17416 

8.17557 
8.17  697 
8.17837 

0     9 

5o 
40 
So 
20 

ID 

52  o 

lO 
20 

3o 

8.17971 
8.18  no 
8.18249 
8.18387 

8.17976 
8.18  ii5 
8.18254 
8.18  392 

0     8 
5o 
4o 
So  7 

I.  Cos.     L.  Cotgr.  1 

1'    t 

,     „ 

L.  Sin.    L.Tang. 

62  3o 

4o 
5o 

8.18  387 
8.18  524 
8.18  662 

8.18392 
8.18  53o 
8.18  667 

So 
20 
10 

53  0 

10 
20 
So 

4o 
5o 

8.18798 
8.18935 
8. 19  071 

8.19  206 
8.19  341 
8. 19  476 

8.18  8o4 
8.18940 

8.19  076 

8.19212 
8.19347 
8.19481 

0    7 

5o 
40 
So 
20 
10 

54  0 

10 
20 

So 

4o 
5o 

8.19610 

8.19  744 
8.19877 

8.20  010 
8.20  i43 
8.20  275 

8.  19  616 

8.19  749 
8.19883 

8.20  016 
8.20  149 
8.20  281 

0    6 

5o 
4o 
So 
20 
10 

55  0 

10 
20 

So 
4o 
5o 

8.20  407 
8.20  538 
8.20  669 

8.20  800 

8.20  930 

8.21  060 

8.20  4i 3 
8.20544 
8.20675 

8.20806 
8.20936 

8.21  066 

0    5 

5o 
4o 
So 

20    , 
10 

56  0 

10 
20 
So 
4o 
5o 

8.21  189 
8.21  319 
8.21  447 

8.21  576 
8.21  703 
8.21  83i 

8.21  195 
8.21  324 
8.21453 

8.21  58i 
8.21  709 
8.21  837 

0    4 

5o 

4o 

So 

20 

10 

57  0 

10 
20 

So 
4o 
5o 

8.21  958 

8.22  085 
8.22  211 
8.22  337 
8.22  463 
8,22  588 

8.21  964 

8.22  091 
8.22  217 

8.22  343 
8.22  469 
8.22  595 

0    3 

5o 

4o 

So 

20 
10 

58  0 

10 
20 
So 
4o 
5o 

8.22  71S 
8.22  838 

8.22  962 

8.23  086 
8.23  210 
8.23  333 

8.22  720 
8.22844 

8.22  968 

8.23  092 
8.23216 
8.23339 

0    2 
5o 

4o 

So 

20 
10 

59  0 

10 
20 
So 
4o 
5o 

8.23  45b 
8.23578 
8.2S  700 

8. 23  822 
8.23  944 
8.24065 

8.23462 
8.23  585 

8. 23  707 
8.23829 
8.23950 

8.24  071 

0    1 

5o 
4o 
So 
20 
10 

60  0 

8.24  186 

8.24  192 

0    0 

L.  Cos. 

L.  Cotg. 

"     ' 

126 


89°. 


FUNCTIONS    OF    SMALL    ANGLES. 
1=. 


/      n 

L.  Sin. 

L.  Tang. 

0    o 

lO 

20 
3o 
40 
5o 

8.24  186 
8.24  3o6 
8.24426 

8.24546 
8.24  665 

8.24785 

8.24  192 
8.243i3 
8.24433 
8.24553 
8.24  672 
8.24  791 

0  60 

5o 
4o 
3o 
20 
10 

1     0 

10 
20 
3o 
4o 
5o 

8.24  903 

8.25  022 
8.25  i4o 

8.25  258 
8.25375 
8.25493 

8.24  910 

8.25  029 
8.25  147 
8.25265 
8.25  382 
8.25  500 

0  59 

5o 
4o 
3o 
20 
10 

2    0 

10 
20 

3o 
4o 
5o 

8.25  609 
8.25  726 
8.25  842 

8.25  958 

8.26  074 
8.26  189 

8.256i6 
8.25  733 

8.25  849 
8.25965 
8.26081 

8.26  196 

0  58 

5o 
4o 

So 
20 
10 

3    0 

10 
20 
So 
4o 
5o 

8.26  So4 
8.26  419 
8.26  533 

8.26  648 
8.26  761 
8.26875 

8.26312 
8.26426 
8.26  541 
8.26655 
8.26  769 
8.26882 

0  57 

5o 
4o 
3o 
20 
10 

4    0 

10 
20 

3o 
4o 
5o 

8.26  988 

8.27  lOI 
8.27  2l4 

8.27326 
8.27438 

8.27  550 

8.26  996 

8.27  109 
8.27  221 

8.27334 
8.27446 
8.27558 

0  66 

5o 
4o 
3o 
20 
10 

6    0 

10 
20 

3o 
4o 
5o 

8.27  Obi 
8.27773 
8.27883 

8.27  994 

8.28  io4 
8.28  215 

8.27  669 
8.27  780 

8.27  891 

8.28  002 
8.28  112 
8.28223 

0  56 

5o 
4o 
So 
20 
10 

6    0 

10 
20 
3o 
4o 
5o 

8.28  324 
8.28434 
8.28543 

8.28  652 
8.28  761 
8.28869 

8.28  332 
8.28442 
8.28  551 

8.28660 
8.28  769 
8.28877 

0  54 

5o 
4o 

So 
20 
10 

0  53 

5o 
4o 

So  52 

7    0 

10 
20 
So 

8.28  977 
8.29085 

8.29  193 
8.29  Soo 

8.28986 
8.29  094 
8.29  201 
8.29  Sog 

L.  Cos. 

L.  Cotg. 

L.  Sin.     L.Tang. 

7  So 

4o 
5o 

8 .  29  Juu 
8.29  407 

8.29  5i4 

8.29  309 
8.29  4i6 
8.29  523 

So 
20 
10 

8    0 

10 
20 

3o 
4o 
5o 

8.29  621 
8.29  727 
8.29833 
8.29  9S9 
8.3oo44 
8.  So  150 

8.29  629 
8.29  736 
8.29  842 
8.29  947 
8.3oo5S 
8. So  i58 

0  52 

5o 
4o 
3o 
20 
10 

9    0 

10 
20 

So 

4o 
5o 

8. So  255 
8. So  359 
8. So  464 
8. So  568 
8. So  672 
8.30776 

8.3o  263 
8. So  368 
8.S047S 
8. So  577 
8. So  681 
8. So  785 

0  51 

5o 
40 

3o 
20 
10 

10  0 

10 
20 
So 
4o 
5o 

8.30879 
8.3098S 
8. Si  086 
8. Si  188 
8.3i  291 
8. Si  S93 

8. So  888 
8. So  992 
8. Si  095 

8. Si  198 
8. 3 1  Soo 
8. Si  4oS 

0  50 

5o 
40 
So 
20 
10 

11   0 

10 
20 
So 
40 
5o 

8. Si  495 
8. Si  597 
8. Si  699 
8. Si  800 
8. Si  901 
8.S2  002 

8.3i  505 
8. Si  606 
8. Si  708 
8. Si  809 
8. Si  911 
8.32  012 

0  49 

5o 
4o 
So 
20 
10 

12  0 

10 
20 

So 
4o 
5o 

8.32  loS 
8.32  20S 
8. 32  3o3 

8.32  4o3 
8. 32  5o3 
8. 32  602 

8.32  112 
8.S2213 
8.323i3 

8.32  4i3 
8.S251S 
8. 32  612 

0  48 

5o 
4o 
So 
20 
10 

13  0 

10 
20 

So 
4o 
5o 

8. 32  702 
8. 32  801 
8.32899 

8. 32  998 

8. 33  096 
8.33  195 

8.32  711 

8. 32  811 
8.S2  909 

8.33  008 
8.33  106 
8.33  205 

0  47 

5o 
4o 
So 
20 
10 

14  0 

10 
20 
So 
4o 
5o 

8.33  292 
8.33  390 
8.33  488 
8.SS585 
8.33  682 
8.33  779 

8.33  302 
8.33  4oo 
8.33498 
8.33595 
8.33692 
8.33  789 

0  46 

5o 
4o 
So 
20 
10 

15  0 

8.33875 

8.33  886 

0  45 

L.  Cos. 

L.  Cotg. 

"     ' 

127 


88°. 


FUNCTIONS    OF    SMALL    ANGLES. 
1°. 


,  „ 

L.  Sin. 

L.  Tang. 

,    „ 

L.  Sin.    L.Tang. 

15  ^> 

lO 

20 
3o 
4o 
5o 

?5.S3  875 
8.33972 
8.34068 

8.34  i64 
8.34260 
8.34355 

8.33  886 
8.33982 
8.34078 

8.34174 

8.34  270 
8.34  366 

0  45 

60 
4o 
3o 

20 

10 

22  3o 

4o 
5o 

8.38  oi4 
8.38  loi 
8.38  189 
8.38  276 
8.38  363 
8.38  45o 

8.38  537 
8.38  624 
8.38710 

8.38026 
8.38  n4 
8.38  202 

3o 
20 
10 

23  0 

10 
20 
3o 
4o 
5o 

8.38289 
8.38376 
8.38  463 
8.38550 
8.38  636 
8.38  723 

0  37 

5o 
4o 
So 
20 
10 

16  0 

10 
20 

3o 
4o 
5o 

8.34450 
8.34546 
8.34  64o 
8.34735 
8.34  83o 

8. 34  924 

8.35  018 
8.35  112 
8.35  206 
8.35  299 
8.35392 
8.35  485 

8.34461 
8.34  556 
8.34  651 

8.34746 
8.34  84o 
8.34935 

0  44 

5o 
4o 
So 
20 
10 

24  0 

10 
20 

So 

4o 
5o 

8.38  796 

8.38  882 
8.38968 

8.S9  o54 

8.39  139 
8.39  226 

8.38  809 
8.38895 

8.38  981 

8.39067 

8.39  i53 
8.S9238 

0  36 

60 
4o 
So 
20 
10 

17  0 

10 
20 
So 

4o 
5o 

8.35  029 
8.35  123 
8.35217 
8.35310 
8.35  4o3 
8.35497 

0  43 

5o 
4o 
3o 
20 
10 

25  0 

10 
20 

So 
4o 
5o 

8.39  3io 
8.39  396 
8.S9480 
8.S9  665 
8.39649 
8.39734 

8.39  S23 
8.S9408 
8.39493 
8.39  578 
8.39663 
8.39747 

0  35 

5o 
4o 
So 
20 
10 

18  0 

10 
20 
So 
4o 
5o 

8.35578 
8.35  67r 
8.35  764 

8.35  856 
8.35948 

8.36  o4o 

8.35  590 

8.35  682 
8.35775 

8.35867 
8.35959 

8.36  o5i 

0  42 

5o 
40 
So 
20 
10 

26  0 

10 
20 

3o 
4o 
60 

8.S9818 

8.39  902 
8.S9986 

8.40  070 
8.4o  i53 
8.40237 

8.39832 
8.39916 
8.40  000 
8.4oo83 
8.40  167 
8.40261 

0  34 

60 
4o 

3o 
20 

10 

19  0 

10 
20 

So 

4o 
5o 

8.36  iSi 
8.36  223 
8.363i4 
8.36  4o5 
8.36496 
8.36  587 

8.36  143 
8.36235 
8.36  326 

8.3641? 
8.36  5o8 
8.36  599 

0  41 

5o 
4o 
So 
20 
10 

27  0 

10 
20 

3o 
4o 
60 

8.40  320 
8.40  4o3 
8.4o486 
8.40669 
8.4o65i 
8.4o  734 

8.4o334 
8.4o4i7 
8.4o  500 

8.4o583 
8.4o665 
8.40748 

0  33 

60 
4o 

So 
20 
10 

20  0 

10 
20 
So 
4o 
5o 

8.36678 
8.36768 
8.36  858 

8.36  948 
8.37038 

8.37  128 

8.36  689 
8.36780 
8.36870 
8.36960 

8.37  050 
8.37  i4o 

0  40 

5o 
4o 
3o 
20 
10 

28  0 

10 
20 

3o 
4o 
5o 

8.40816 
8.40898 
8.40980 

8.4i  062 
8.4i  i44 
8.4i  226 

8.4o83o 
8.40913 
8.40995 

8.4i  077 
8.4i  168 
8.4i  24o 

0  32 

60 
4o 
So 
20 
10 

21  0 

10 

20 
So 
4o 
5o 

8.37  217 
8.37306 
8.37395 

8.37484 
8.37673 
8.37662 

8.37  229 
8.37318 
8.37408 

8.S7497 
8.37  585 
8.37674 

0  39 

5o 
4o 
So 
20 
10 

29  0 

10 
20 
So 
40 
60 

8.4i  307 
8.4i  388 
8.4i  469 
8.4i  55o 
8.4i  63i 
8.4i  711 

8.4i  321 
8.41  4o3 
8.4i484 
8.4i  565 
8.4i  646 
8.41  726 

0  31 

60 
4o 
3o 
20 
10 

22  0 

10 
20 
3o 

8.37750 
8.37838 
8.37926 

8.38oi4 

8.37762 
8.37860 
8.37938 
8.38026 

0  38 

60 
4o 

So  37 

80  0 

8.4i  792 

8. 4 1  807 

0  80 

.  L.  Cos. 

L.  Cotg. 

/,    / 

L.  Cos. 

L.  Cotg. 

'     " 

128 


88°. 


FUNCTIONS    or    SMALL    ANGLES. 

1°. 


,        n 

L.  Sin.    L.Tang. 

30  0 

lO 
20 

3o 
4o 
5o 

8.4i  792 
8.4i  872 
8.4i  952 
8.42032 
8.42  112 
8.42  192 

8.4i  807 
8.4i  887 
8.4i  967 
8.42  o48 

8.42   12" 

8.42  207 

0  30 

5o 
4o 
3o 
20 
10 

31  o 

lO 
20 

3o 
4o 
5o 

8.42  272 
8.42  351 
8.42430 
8.42  5io 
8.42589 
8.42667 

8.42  207 
8.42  366 
8.42  446 

8.42  525 
8.42  6o4 
8.42  683 

0  29 

5o 
4o 
3o 
20 
10 

32  o 

lO 
20 

3o 
4o 
5o 

8.42  746 
8,42825 
8.42  903 

8.42  982 
8.43060 

8.43  i38 

8.42  762 
8.42840 
8.42  919 

8.42  997 
8.43075 

8.43  i54 

0  28 

5o 
4o 
3o 
20 
10 

33  o 

lO 

20 

3o 
4o 
5o 

8.43  216 
8.43293 
8.43371 

8.43448 
8.43  526 
8.436o3 

8.43  232 
8.43  309 
8.43387 

8.43  464 
8,43542 
8.43619 

0  27 

5o 
4o 

3o 
20 
10 

34  o 

lO 
20 

3o 
4o 
5o 

a.4-i  680 
8.43757 
8.43  834 

8.43  910 
8.43987 

8.44  o63 

8.43  696 
8.43773 
8.43  85o 

8.43  927 

8.44  oo3 
8.44080 

0  26 

5o 
4o 

3o 
20 
10 

35  o 

lO 
20 

3o 

4o 
5o 

8.44 139 
8.44216 
8.44  292 
8.44367 
8.44443 
8.445rq 

8.44  i56 

8.44  232 

8.44308 

8.44  384 
8.44460 
8.44  536 

0  25 

5o 
4o 
3o 
20 
10 

36  o 

lO 
20 

3o 
4o 
5o 

8.44  594 
8.44669 
8.44745 
8.44820 
8.44895 
8.44969 

8.44611 
8.44  686 
8.44  762 
8,44  837 
8.44  912 
8,44987 

0  24 

5o 
4o 
3o 
20 
10 

37  o 

lO 
20 

3o 

8.45  o44 
8.45  119 
8.45  193 
8.45267 

8.45061 
8,45  i36 
8.45  210 

8.45  285 

0  23 

5o 

4o 

3o22 

L.  Cos.     L.  Cotg. 

"     ' 

'     '/ 

L.  Sin.   ;  L.Tang. 

37  3o 
4o 
5o 

8.45  267 
8.45341 
8.454i5 

8.45  285 
8.45359 
8.45433 

3o 
20 
10 

38  0 

10 
20 
3o 
4o 
5o 

8.45  489 
8.45  563 
8.45  637 
8.45710 
8.45784 
8.45  857 

8.45  507 
8.45  581 
8.45  655 

8.45728 
8.45802 
8.45875 

0  22 

5o 
4o 
3o 
20 
10 

39  0 

10 
20 

3o 
4o 
5o 

8.45930 
8,46  oo3 
8.46  076 

8.46  149 
8,46  222 
8.46294 

8.45  948 

8.46  021 
8.46094 
8.46  167 
8.46240 
8.46  312 

0  21 

5o 
4o 

3o 
20 
10 

40  0 

10 
20 

3o 
4o 
5o 

8.46  366 
8.46439 
8.465II 

8.46  583 
8.46  655 
8,46  727 

8.46  385 
8,46  45? 
8,46  529 

8.46602 
8.46674 
8.46745 

0  20 

5o 

4o 

3o 
20 
10 

41  0 

10 
20 

3o 
4o 
5o 

8.46  799 
8.46870 
8,46942 

8.47  oi3 
8.47084 
8.47  1 55 

8,46817 
8,46889 
8,46960 

8,47  o32 
8,47  io3 
8,47  174 

0  19 

5o 
4o 
3o 
20 
10 

42  0 

10 
20 
3o 
4o 
5o 

8,47  226 
8,47  297 
8.47  368 

8.47439 
8.47  509 
8.47580 

8,47245 
8.47316 
8,47387 
8,47  458 
8.47528 
8.47  599 

0  18 

5o 

4o 

3o 
20 
ro 

43  0 

10 
20 

3o 
4o 
5o 

8.47  650 
8.47  720 
8.47  790 
8.47860 
8.47  930 
8.48000 

8,47669 
8,47  740 
8.47810 

8,47  880 

8.47950 
8.48020 

0  17 

5o 

4o 
3o 
20 
10 

44  0 
10 
20 

3o 

4o 
5o 

8.48069 
8.48  139 
8.48208 

8,48  278 
8.48  347 
8.484i6 

8,48090 
8.48  i59 
8,48228 

8,48298 
8,48  367 
8,48  436 

0  16 

5o 
4o 
3o 
20 
10 

0  15 

45  0 

8.48485 

8.48  5o5 

L.  Cos.     L.  Cotg. 

"     ' 

88°. 


129 


FUNCTIONS    OF    SMALL   ANGLES. 

1°. 


,  „ 

L.  Sin. 

L.  Tang. 

,       „ 

L.Sin.  iL.Tang. 

45  0 

lO 
20 

3o 
4o 
5o 

8.48  485 
8.48  554 
8.48  622 

8.48691 
8.48760 
8.48828 

8.48  5o5 
8.48  574 
8.48  643 

8.48  711 
8.48780 
8.48849 

0  15 

5o 
4o 
3o 
20 
10 

52  3o 
4o 
5o 

8.5i  480 
8.5i  544 
8. 5 1  609 

8.5i  5o3 
8.5i  568 
8.5i  632 

3o 
20 
10 

53  0 

10 
20 
3o 
4o 
5o 

8.5i  673 
8.5i  737 
8.5i  801 

8.5i  864 
8.5i  928 
8.5i  992 

8.5i  696 
8,5i  760 
8,5i  824 
8.5i  888 
8.5i  952 
8.520I5 

0    7 

5o 
4o 
3o 
20 
10 

46  o 

lO 
20 

3o 
4o 
5o 

8.48896 
8.48965 
8.49033 

8.49  lOI 
8.49  169 
8.49  236 

8.48917 
8.48985 
8.49  o53 
8.49  121 
8.49  189 
8.49  257 

0  14 

5o 
4o 
3o 
20 
10 

54  0 

10 
20 
3o 
4o 
5o 

8.52  o55 
8.52  119 
8.52  182 

8.52245 
8.52  3o8 
8.52371 

8.52  079 
8,52  143 
8,52  206 
8,52  269 
8.52  332 
8,52896 

0    6 

5o 
4o 
80 
20 
10 

0    5 

5o 

4o 

3o 
20 
10 

47  o 

10 
20 

3o 
4o 
5o 

8.49  3o4 
8.49372 
8.49439 
8.49  5o6 
8.49574 
8.49641 

8.49325 
8.49393 
8.49460 
8.49528 
8.49595 
8.49662 

0  13 

5o 
40 
3o 
20 
10 

55  0 

10 
20 
3o 

4o 
5o 

8.52434 
8.52497 
8.52  56o 

8.52623 
8.52  685 
8.52748 

8.52459 

8.52  522 

8.52  584 

8.52647 
8.52  710 
8,52  772 

48  o 

lO 
20 

3o 
4o 
5o 

8.49  708 

8.49775 
8.49842 

8.49908 

8.49  975 

8.50  042 

8.49  729 
8.49  796 
8.49863 

8.49  980 
8.49997 
8.5oo63 

0  12 

5o 
4o 

3o 
20 
10 

0  11 

5o 
4o 
3o 
20 
10 

56  0 

10 
20 
3o 
40 
5o 

8,52810 
8,52  872 
8.52  935 

8.52  997 

8.53  059 
8.53  121 

8.52  885 
8.52897 

8.52  960 

8.53  022 
8,53  o84 
8.53  i46 

0    4 

5o 
4o 
3o 
20 
10 

49  o 

10 
20 

3o 
4o 
5o 

8.5o  108 
8.5o  174 
8.5o  241 
8.5o  307 
8.50873 
8.50439 

8.5o  i3o 
8.5o  196 
8.50  263 

8.5o  329 
8.5o395 
8.5o46i 

57  0 

JO 

20 
So 
4o 
5o 

8.53  i83 
8.53245 
8,53  3o6 

8,53  368 
8,53429 
8,53491 

8,58208 
8.53  270 
8.53  332 
8.53393 
8.53455 
8.535i6 

0    3 

5o 
4o 

3o 
20 

TO 

50  o 

lO 
20 

3o 
4o 
5o 

8.5o5o4 
8.5o  570 
8.5o636 

8.5o  701 
8.5o  767 
8.5o832 

8.5o  527 
8.50593 
8.5o658 

8.5o  724 
8.5o  789 
8.5o855 

0  10 

5o 
4o 
3o 
20 
10 

58  0 

10 
20 
3o 
4o 
5o 

8.53  552 
8,536i4 
8,53675 

8.53786 
8.53  797 
8.53  858 

8.53578 
8.53689 
8.53  700 

8,58762 
8,53  823 
8.58  884 

0    2 

5o 
4o 
80 
20 

ID 

51  o 

ID 
20 

3o 
4o 
5o 

8.50  897 
8.50963 
8.5i  028 

8. 5 1  092 
8.5i  i57 
8.5i  222 

8.50  920 
8.50985 
8.5i  o5o 

8.5i  ii5 

8. 5 1  180 
8.5i  245 

0     9 

5o 
4o 
3o 
20 
10 

59  0 

10 
20 

3o 
4o' 
5o 

8.53919 

8.53  979 

8.54  o4o 

8,54  101 
8,54  161 
8,54  222 

8,58945 
8.54  oo5 
8,54066 

8,54  127 
8,54  187 
8,54  248 

0       1 

5o 
4o 
3o 
20 
10 

52  o 

lO 
20 

3o 

8.5i  287 
8.5i  35i 
8.5i4i6 
8.5i  480 

8.5i  3io 
8.5i  374 
8.5i  439 
8.5i  5o3 

0     8 

5o 
4o 
3o    7 

60  0 

8.54282 

8.54308 

0    0 

L.  Cos. 

L.  Cotg. 

n       ' 

L.  Cos. 

L.  Cotg. 

"    ' 

130 


88°. 


TABLE    IV 

FOUR-PLACE 
NAPERIAN    LOGARITHMS 


NAPERIAN    LOGARITHMS. 

LOGARITHMS    OF    POWERS    OF    lo. 


Num. 

Log. 

lO 

2.3026 

lOO 

4.6o52 

lOOO 

6.9078 

lOOOO 

9.2108 

lOOOOO 

1 1 .5129 

lOOOOOO 

i3.8i55 

lOOOOOOO 

16.U81 

lOOOOOOOO 

18.4207 

lOOOOOOOOO 

20.7233 

Num. 

Log. 

Num. 

Log. 

.  I 

3.6974 

.01 

5.3948 

.001 

7.0922 

.000 1 

10.7897 

.00001 

12.4871 

.000001 

14. 1845 

.0000001 

77.8819 

.0000000 I 

19.6793 

.000000001 

21 ,2767 

Num. 

Log. 

LOGARITHMS  OF  NUMBERS  FROM  i  TO  10. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.0 

I .  I 
1 .2 
1.3 

1.4 
1.5 
1.6 

1-7 
1.8 
1.9 

2.0 

0.0000 

0100 

0198 

0296 

0892 

o4B8 

o588 

0677 

0770 

0862 

0.0953 
0.1823 
0.2624 

0.3365 
o.4o55 
0.4700 

o.53o6 
0.5878 
0.6419 

io44 
1906 
2700 

3436 

4l2I 

4762 

5365 
5933 
647' 

ii33 
1989 
2776 

3507 
4187 
4824 

5423 
5988 
6523 

1222 
2070 
2862 

3577 
4253 
4886 

5481 
6043 
6675 

i3io 

2l5l 

2927 

8646 
43i8 
4947 

5539 
6098 
6627 

1898 
2281 
8001 

3716 
4388 
5oo8 

5596 
6i52 
6678 

i484 

23ll 

8075 

8784 
4447 
5o68 

5658 
6206 
6729 

1670 
2890 
3i48 

8853 
45ii 
5i28 

5710 
6269 
6780 

i655 
2469 

3221 

8920 

4574 

5i88 

5766 
63i3 
6881 

1740 
2546 
8298 

8988 
4637 
5247 

5822 

6366 

6881 

0.6981 

6981 

7o3i 

7080 

7129 

7178 

7227 

7275 

7324 

7872 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

182 


NAPERIAN 

LOGARITHMS 

N 

0 

1 

2 

3 

4 

5  i  6  1 

7 

8    9 

2.0 

2.t 
2.2 
2.3 

2.4 
2.5 
2.6 

2.7 
2.8 
2.9 

3.0 

3.1 

3.2 

3.3 

3.4 
3.5 
3.6 

3.7 
3.8 

3.9 
4.0 

4.1 

4.2 

4.3 

4.4 
4.5 
4.6 

4.7 
4.8 

4.9 
5.0 

5.1 

5.2 

5.3 

5.4 
5.5 
5.6 

5.7 
5.8 
5.9 

6.0 

0.6931 

6981 

7o3i 

7080 

7129 

7178 

7227 

7275 

7324  7372 

0.7419 
0.7885 
0.8329 

0.8755 
0.9163 
0.9555 

0.9933 
1.0296 
1.0647 

7467 
7930 
8372 

8796 
9203 
9594 

9969 
o332 
0682 

75i4 
7975 
84i6 

8838 
9243 
9632 

6006 
0367 
0716 

756i 
8020 
8459 

8879 
9282 
9670 

0043 
o4o3 
0750 

7608 
8o65 
85o2 

8920 
9322 
9708 

0080 
o438 
0784 

7655 
8109 

8544 

8961 
9361 
9746 

6116 

0473 
0818 

7701 
8i54 
8587 

9002 
9400 
9783 

6i52 
o5o8 
o852 

7747 
8198 
8629 

9042 
9439 
9821 

0188 
o543 
0886 

7793 
8242 
8671 

9083 
9478 
9858 

6225 
0578 
0919 

7»39 
8286 
8713 

9123 
9517 
9895 

0260 
06 1 3 
0953 

1 .0986 

1019 

io53 

1086 

i4io 
1725 
2o3o 

2326 
2613 

2892 

3i64 
3429 
3686 

1 1 19 

I  i5i 

ii84 

1217 

1249 

1282 

i.i3i4 
i.i632 
I. 1939 

1.2238 

1.2528 
1.2809 

i.3o83 
i.335o 
I.36IO 

1 346 
i663 
1969 

2267 
2556 
2837 

3iio 
3376 
3635 

1378 
1694 
2000 

2296 
2585 
2865 

3i37 
34o3 
366i 

i442 
1756 
2060 

2355 
2641 
2920 

3191 
3455 
3712 

1474 
1787 
2090 

2384 
2669 
2947 

3218 
3481 
3737 

i5o6 
1817 
2119 

24l3 

2698 
2975 

3244 
3507 
3762 

1537 
1 848 
2149 

2442 
2726 
3oo2 

3271 

3533 
3788 

1069 
1878 
2179 

2470 
2754 
3029 

3297 
3558 
38i3 

1600 
1909 
2208 

2499 
2782 
3o56 

3324 
3584 
3838 

1.3863 

3888 

3913 

3938 

3962 

3987 

4oi2 

4o36 

4o6i 

4o85 

1.4110 
i.435i 
1.4586 

1.4816 
i.5o4i 
1.5261 

1.5476 
1.5686 
1.5892 

4i34 
4375 
4609 

4839 
5o63 
5282 

5497 
5707 
5913 

6ii4 

4i59 
4398 
4633 

486 1 
5o85 
53o4 

55i8 
5728 
5933 

4i83 
4422 
4656 

4884 
5i07 
5326 

5539 
5748 
5953 

4207 
4446 
4679 

4907 
5129 
5347 

556o 
5769 
5974 

423i 
4469 
4702 

4929 
5i5i 
5369 

558i 
5790 
5994 

4255 
4493 
4725 

4951 
5173 
5390 

56o2 
58io 
60 1 4 

4279 
45i6 
4748 

4974 
5195 
5412 

5623 
583i 
6o34 

43o3 
4540 
4770 

4996 
5217 
5433 

5644 
585i 
6o54 

4327 
4563 
4793 

5019 
5239 
5454 

5665 
5872 
6074 

1 .6094 

6i34 

6i54 

6174 

6194 

6214 

6233 

6429 
6620 
6808 

6993 
7174 
7352 

7527 
7699 
7867 

6253 

6273 

1 .6292 
1.6487 
1.6677 

1.6864 
1.7047 
1.7228 

i.74o5 
1.7579 
1.7750 

63i2 
65o6 
6696 

6882 
7066 
7246 

7422 
7596 
7766 

6332 
6525 
6715 

6901 
7084 
7263 

7440 
7613 
7783 

635i 
6544 
6734 

6919 
7102 
7281 

7457 
7630 
7800 

6371 
6563 
6752 

6938 
7120 
7299 

7475 
7647 
7817 

6390 
6582 
6771 

6956 
7i38 
7317 

7492 
7664 
7834 

6409 
6601 
6790 

6974 
7i56 
7334 

7509 
7681 
785i 

6448 
6639 
6827 

701 1 
7192 
7370 

7544 
7716 
7884 

6467 
6658 
6845 

7029 
7210 
7387 

7561 
7733 
7901 

1. 7918 

7934 

7951 

7967 

7984 

8001 

8017 

8o34 

8o5o 

8066 

0 

1 

2  1  3 

4 

5 

6 

7 

8 

9 

i33 


NAPERIAN 

LOGARITHMS 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

6.0 

6.1 

6.2 

6.3 

1 .7918 

7934 

7951 

7967 

7984 

8001 

8017 

8o34 

8o5o 

8066 

i.8o83 
1.8245 
i.84o5 

8099 
8262 
8421 

8116 

8278 
8437 

8i32 
8294 
8453 

8i48 
83io 
8469 

8i65 
8326 
8485 

8181 
8342 
85oo 

8197 
8358 
85i6 

8213 
8374 
8532 

8229 
8390 
8547 

6.4 
6.5 
6.6 

1.8563 
I. 8718 
I. 8871 

8579 
8733 
8886 

8594 

8749 
8901 

8610 
8764 
8916 

8625 
8779 
8931 

864i 
8795 
8946 

8656 
8810 
8961 

8672 
8825 
8976 

8687 
884o 
8991 

8703 
8856 
9006 

6.7 
6.8 
6.9 

7.0 

7-1 
7.2 
7.3 

1 .9021 
1 .9169 
1.9315 

9o36 
9184 
9330 

905 1 
9199 

9344 

9066 
9213 
9359 

9081 
9228 
9373 

9095 
9242 
9387 

9110 
9257 
9402 

9125 
9272 
9416 

9140 
9286 
943o 

9155 
9301 
9445 

1.9459 

9473 

9488 

9502 

9516 

9530 

9544 

9559 

9573 

9587 

I .9601 

1.9741 
1.9879 

9615 
9755 
9892 

9629 
9769 
9906 

9643 
9782 
9920 

9657 
9796 
9933 

9671 
9810 
9947 

9685 
9824 
9961 

9699 
9838 
9974 

9713 
9851 
9988 

9727 
9865 
0001 

7-4 
7.5 
7.6 

2.00l5 

2.0149 
2.0281 

0028 
0162 
0295 

0042 
0176 
o3o8 

oo55 
0189 

o321 

0069 
0202 
o334 

0082 
021 5 
o347 

0096 
0229 
o36o 

0109 
0242 
0373 

0122 
0255 
o386 

oi36 
0268 
0399 

7-7 
7.8 

7-9 

8.0 

8.1 
8.2 
8.3 

2.o4l2 

2.o54i 

2.0669 

o425 
o554 
0681 

o438 
0567 
0694 

045 1 
o58o 
0707 

o464 
0592 
0719 

0477 
o6o5 
0732 

0490 
0618 
0744 

o5o3 
o63i 
0757 

o5i6 
0643 
0769 

o528 
o656 

0782 

2.0794 

0807 

0819 

o832 

0844 

0857 
0980 

I102 
1223 

0869 

0882 

0894 

0906 

2.0919 
2. 104l 

2.1163 

0931 
io54 
1 175 

0943 
1066 

1187 

0956 
1078 
1199 

0968 
1090 
1211 

0992 
11 14 
1235 

ioo5 
1 126 

1247 

1017 
ii38 
1258 

1029 
II  5o 
1270 

8.4 
8.5 
8.6 

2. 1282 
2.  l4oi 

2.i5i8 

1294 

l4l2 

1529 

i3o6 

1424 
i54i 

i3i8 
i436 
i552 

i33o 
1 448 
i564 

I  342 
1459 
1576 

i353 
1471 
i587 

i365 
i483 
1599 

1377 
1494 
1610 

1389 
i5o6 
1622 

8.7 
8.8 
8.9 

9.0 

9.1 
9.2 
9.3 

2.1633 

2.1748 
2.1861 

1645 
1759 
1872 

i656 
1770 
i883 

1668 
1782 
1894 

1679 
1793 
1905 

169I 

i8o4 
1917 

1702 
i8i5 
1928 

1713 
1827 
1939 

1725 
i838 
1950 

1736 
1849 
1961 

2. 1972 

1983 

1994 

2006 

2017 

2028 

2039 

2o5o 

2061 

2072 

2.2083 

2.2192 

2.2300 

2094 

22o3 
23ll 

2io5 

22l4 
2^2 

2116 

2225 
2332 

2127 

2235 

2343 

2i38 

2246 

2354 

2i48 
2257 
2364 

2159 

2268 
2375 

2170 
2279 
2386 

2181 
2289 
2396 

9.4 
9.5 
9.6 

2.2407 

2.25l3 

2.2618 

2418 
2523 

2628 

2428 

2534 
2638 

2439 

2544 
2649 

245o 
2555 
2659 

2460 
2565 
2670 

2471 
2576 
2680 

2481 

2586 
2690 

2492 
2597 
2701 

25o2 
2607 
271  1 

9-7 
9.8 
9.9 

10.0 

2.2721 
2.2824 
2.2925 

2732 
2834 
2935 

2742 
2844 
2946 

2752 
2854 
2956 

2762 
2865 
2966 

2773 
2875 
2976 

2783 
2885 
2986 

2793 
2895 
2996 

28o3 
2905 
3oo6 

2814 
2915 

3oi6 

2.3026 

3126 

3224 

3322 

34i8 

35i4 

3609 

3703 

3796 

3888 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 34 


TABLE    V 

FOUR-PLACE    LOGARITHMS 
OF    NUMBERS 


FOUR-PLACE    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

1 1 

0000 

043 

086 

128 

170 

212 

253 

294 

334 

374 

4i4 

453 

492 

53i 

569 

607 

645 

682 

719 

755 

12 

792 

828 

864 

899 

934 

969 

ioo4 

io38 

1072 

1106 

i3 

1139 

173 

206 

239 

271 

3o3 

335 

367 

399 

43o 

i4 

46i 

492 

523 

553 

584 

6i4 

644 

673 

703 

782 

i5 

1 76 1 

790 

818 

847 

875 

903 

981 

959 

987 

20l4 

i6 

2o4l 

068 

095 

122 

i48 

175 

20I 

227 

253 

279 

17 

3o4 

33o 

355 

38o 

4o5 

43o 

455 

48o 

5o4 

529 

18 

553 

577 

601 

625 

648 

672 

695 

7.8 

742 

765 

J9 
20 

21 

788 

810 

833 

856 

878 

900 

923 

945 

967 

989 

3oio 

o32 

o54 

075 

096 

118 

189 

160 

181 

201 

222 

243 

263 

284 

3o4 

324 

345 

•365 

385 

4o4 

22 

424 

444 

464 

483 

5o2 

522 

541 

56o 

579 

598 

23 

617 

636 

655 

674 

692 

711 

729 

747 

766 

784 

24 

802 

820 

838 

856 

874 

892 

909 

927 

945 

962 

25 

3979 

997 

4oi4 

4o3i 

4o48 

4o65 

4082 

4099 

4ii6 

4 1 33 

26 

4i5o 

166 

i83 

200 

216 

232 

249 

265 

281 

298 

27 

3i4 

33o 

346 

862 

378 

393 

409 

425 

44o 

456 

28 

472 

487 

5o2 

5i8 

533 

548 

564 

579 

594 

609 

29 

30 

3i 

624 

639 

654 

669 

683 

698 

7i3 

728 

742 

757 

4771 

786 

800 

8i4 

829 

843 

857 

87, 

886 

900 

914 

928 

942 

955 

969 

983 

997 

5oi  I 

5o24 

5o38 

32 

5o5r 

o65 

079 

092 

io5 

"9 

l32 

145 

159 

172 

33 

i85 

198 

211 

224 

237 

25o 

263 

276 

289 

802 

34 

3i5 

328 

34o 

353 

366 

378 

391 

4o3 

4i6 

428 

35 

5441 

453 

465 

478 

490 

5o2 

5i4 

527 

539 

55i 

36 

563 

575 

587 

599 

611 

623 

635 

647 

658 

670 

37 

682 

694 

7o5 

717 

729 

74o 

752 

763 

775 

786 

38 

798 

809 

821 

832 

843 

855 

866 

877 

888 

899 

39 
40 

911 

922 

933 

944 

955 

966 

977 

988 

999 

6010 

6021 

o3i 

042 

o53 

064   075   o85  1 

096 

107 

117 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

.1 

38 

32 

38 

as 

.1 

33 

31 

»9 

.1 

18 

17 

16 

3.8 

32 

2.8 

2-S 

2.2 

2.1 

1.9 

1.8 

»-7 

1.6 

.2 

76 

6.4 

5.6 

5.0 

.2 

4-4 

4-2 

3-8 

.2 

S-b 

3-4 

3o 

•3 

II. 4 

9.6 

8.4 

7-5 

•3 

6.6 

6-3 

5-7 

•  3 

S-4 

S> 

4.8 

•4 

IS-2 

r2  8 

II. 2 

lao 

•4 

8.8 

8.4 

7.6 

•4 

7.2 

6.8 

6.4 

•5 

iq.o 

16.0 

14.0 

12.5 

•5 

II. 0 

10.5 

9-5 

•  s 

9.0 

8-5 

8.0 

.6 

22.8 

19.2 

16.8 

ISO 

.6 

13.3 

13.  b 

11.4 

.6 

10.8 

ia3 

9.6 

•7 

26.6 

22.4 

19.6 

17-5 

•7 

I7.6 

«4.7 

•3-3 

.7 

12.6 

11.9 

II. 2 

.8 

30.4 

25.6 

22.4 

20.0 

.8 

16.8 

15.2 

.8 

14.4 

13.6 

12.8 

.9    34.2  1  28.8  1  25.2  1  22.5  1 

wm^mmm 

19.8    18.9  1  17. 1  1 

.9    i6.2  1  15.3  1  14.4  1 

1 36 


FOUR-PLACE    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

40 

4i 

6021 

o3i  1  042 

o53 

064 

075 

o85 

096 

107 

117 

128 

i38 

149 

160 

170 

180 

191 

201 

212 

222 

42 

232 

243 

253 

263 

274 

284 

294 

3o4 

3i4 

325 

43 

335 

345 

355 

365 

375 

385 

395 

4o5 

4i5 

425 

44 

435 

444 

454 

464 

474 

484 

493 

5o3 

5i3 

522 

45 

6532 

542 

55i 

56i 

■571 

58o 

590 

599 

609 

618 

46 

628 

637 

646 

656 

665 

675 

684 

693 

702 

712 

47 

721 

73o 

739 

749 

758 

767 

776 

785 

794 

8o3 

48 

812 

821 

83o 

839 

848 

857 

866 

875 

884 

893 

49 
50 

5i 

902 

911 

920 

928 

937 

946 

955 

964 

972 

981 

6990 

998 

7007 

7016 

7024 

7033 

7042 

7o5o 

7059 

7067 

7076 

084 

093 

lOI 

no 

118 

126 

i35 

143 

l52 

52 

160 

168 

177 

i85 

193 

202 

210 

218 

226 

235 

53 

243 

25l 

259 

267 

275 

284 

292 

3oo 

3o8 

3i6 

54 

824 

332 

34o 

348 

356 

364 

372 

38o 

388 

396 

55 

74o4 

4l2 

419 

427 

435 

443 

45i 

459 

466 

474 

56 

482 

490 

497 

5o5 

5i3 

520 

528 

536 

543 

55i 

5? 

559 

566 

574 

582 

589 

597 

6o4 

612 

619 

627 

58 

634 

642 

649 

657 

664 

672 

679 

686 

694 

701 

i>9 
60 

6i 

709 

716 

723 

73i 

738 

745 

752 

760 

767 

774 

7782 

789 

796 

8o3 

810 

818 

825 

832 

839 

846 

853 

860 

868 

875 

882 

889 

896 

.  903 

910 

917 

62 

924 

931 

938 

945 

952 

959 

966 

973 

980 

987 

63 

993 

8000 

8007 

8oi4 

8021 

8028 

8o35 

804 1 

8o48, 

8o55 

64 

8062 

069 

075 

082 

089 

096 

102 

109 

116 

122 

65 

8129 

i36 

1 42 

149 

i56 

162 

169 

176 

182 

189 

66 

195 

202 

209 

2l5 

222 

228 

235 

241 

248 

254 

67 

261 

267 

274 

280 

287 

293 

299 

3o6 

3l2 

319 

68 

325 

33i 

338 

344 

35i 

357 

363 

370 

376 

382 

69 
70 

388 

395 

4or 

407 

4i4 

420 

426 

432 

439 

445 

45r 

457 

463 

470 

476 

482 

488 

494 

5oo 

5o6 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

15 

14 

13 

13 

.1 

II 

10 

9 

.1 

8 

7 

6 

i.S 

1.4 

1-3 

1.2 

I.I 

I.O 

0.9 

0.8 

0.7 

0.6 

.2 

3.0 

z.t 

2.6 

2.4 

.2 

2.2 

2.0 

1.8 

.2 

1.6 

1.4 

1.2 

•3 

4-5 

4.2 

3-9 

3.b 

•3 

3-3 

30 

2.7 

•3 

2.4 

2.1 

1.8 

•4 

6.0 

5C 

5-2 

4.8 

•4 

4-4 

4.0 

3.6 

•4 

3-2 

2.8 

2.4 

•5 

7-5 

7<: 

>   6.S 

6.0 

•5 

S-S 

■S-o 

45 

•  5 

4,0 

3-5 

3.0 

.6 

9.0 

8.4 

7.8 

7.2 

.6 

6.6 

b.o 

5-4 

.6 

4.8 

4.2 

3-6 

•7 

10.5 

9.  J 

9.. 

8.4 

•7 

7-7 

7.0 

6.3 

•7 

S-6 

4.0 

4.2 

.8 

12.0 

II. 2 

10.4 

9.6 

.8 

8.8 

8,0 

7.2 

8 

b.4 

5-6 

48 

.9    13.S  1  12  6  1  117  1  T0.8 

9.9    9.0  1  8.1 

.9    7.2  1  6.3 

5-4 

137 


FOUR-PLACE 

LOGARITHMS. 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

70 

71 
72 

73 

845i 

457 

463 

470 

476 

482 

488 

494 

5oo 

5o6 

5i3 
573 
633 

519 

579 
639 

525 

585 
645 

53i 
59, 
65i 

537 
597 
657 

543 
6o3 
663 

549 
609 
669 

555 
6i5 
675 

56i 
621 
681 

567 
627 
686 

74 
75 
76 

692 

8751 

808 

698 
756 
8i4 

704 
762 
820 

710 
768 
825 

716 

774 
83i 

722 

779 
837 

727 
785 

842 

733 
791 
848 

739 

797 
854 

745 
802 
859 

77 

78 

79 

80 

81 
82 
83 

865 
921 
976 

871 
927 
982 

o36 

876 
932 

987 

882 
938 
993 

887 
943 
998 

893 

949 
9004 

899 
954 

9009 

904 
960 
9015 

910 
965 
9020 

915 
971 
9025 

9o3i 

o42 

o47 

lOI 

1 54 
206 

o53 

o58 

o63 

069 

074 

079 

i33 
186 
238 

o85 
1 38 
191 

090 
i43 
196 

096 
149 
201 

106 
159 
212 

112 
i65 
217 

117 
170 
222 

122 
175 
227 

128 
180 

232 

84 
85 
86 

243 

9294 

345 

248 
299 
35o 

253 
3o4 
355 

258 
309 
36o 

263 
3i5 
365 

269 
820 
370 

274 
325 
375 

279 
33o 
38o 

284 

335 
385 

289 
34o 
390 

87 
88 
89 

90 

91 
92 
93 

395 

445 
494 

4oo 
45o 
499 

4o5 
455 
5o4 

4io 
46o 
509 

4i5 
465 
5i3 

420 

469 

5i8 

425 
474 
523 

43o 

479 
528 

435 
484 
533 

44o 
489 
538 

9542 

547 

552 

557 

562 

566 

57. 

576 

624 
671 
717 

58i 

586 

633 
680 
727 

690 
638 
685 

595 
643 
689 

600 

647 
694 

6o5 
652 
699 

609 

657 
703 

6i4 
661 
708 

619 
666 
7.3 

628 
675 
722 

94 
95 
96 

73i 

9777 
823 

736 
782 
827 

741 
786 
832 

745 
791 
836 

75o 
795 

84i 

754 
800 
845 

759 
8o5 
85o 

763 
809 

854 

768 
8i4 
859 

773 
818 
863 

97 
98 

99 

100 

868 
912 
956 

872 
917 
961 

877 
921 
965 

881 
926 
969 

886 
930 
974 

890 
934 

978 

894 

939 
983 

899 
943 
987 

903 
948 
991 

908 
952 
996 

0000 

oo4 

009 

oi3 

017 

022 

026 

o3o 

o35 

o4o 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

P 

7 

6 

.1 
.2 

■3 

5 

4 

0.7 
'4 
2.1 

0.6 
1.2 

1.8 

0.5 

I.O 

1-5 

0.4 
0.8 
1.2 

2.8 
3-5 
4.2 

2-4 

3-6 

■4 
•3 

.6 

a  0 

2-5 

3.0 

1.6 
2.0 

2-4 

? 

5-6 
6-3 

42 
4.8 

5-4 

•7 
.8 

■9 

3-5 
4.0 
4-5 

2.8 
3.6 

i38 


TABLE    VI 

FOUR-PLACE    LOGARITHMS 

OF   THE 

TRIGONOMETRIC    FUNCTIONS 

TO   EVERY  TEN    MINUTES 


FOUR-PLACE    LOGARITHMIC   FUNCTIONS. 

O        ' 

L.  Sin. 

d. 

L.Tang. 

d.    L.  Cotg. 

L.  Cos. 

d. 

0    o 

0.0000 

0    90 

lO 

7-4637 

7.4637 

2.5363 

0. 

0000 

5o 

20 

7.7648 

1760 

7.7648 

1761 

2.2352 

0. 

0000 

0 

4o 

3o 

7.9408 

7 . 9409 

1249 

2.0591 

0. 

0000 

3o 

4o 

8.0658 

8.0658 

969 

1.9342 

0. 

0000 

20 

5o 

8.1627 

969 

8.1627 

1.8373 

0. 

0000 

° 

1 0 

1     0 

8.2419 

669 

8.2419 

670 

1.7581 

9- 

9999 

I 

0    89 

10 

8.3o88 

580 

8.3089 

580 

I. 691  I 

0. 

9999 

5o 

20 

8.3668 

8.3669 

1. 633 1 

9- 

9999 

4o 

5" 

5" 

3o 

8.4179 

458 

8.4181 

457 

I. 5819 

9- 

9999 

3o 

4o 

8.4637 

8.4638 

1.5362 

9- 

999^ 

20 

5o 

8.5o5o 

4'3 
378 
348 

8.5o53 

415 

378 
348 

1.4947 

9- 

999S 

0 

0 

10 

2    0 

8.5428 

8.5431 

1.4569 

9- 

9997 

0    88 

10 

8.5776 

8.5779 

I .4221 

9- 

9997 

5o 

20 

8.6097 

321 
300 

8.6101 

300 

1.3899 

9- 

9996 

0 

4o 

3o 

8.6397 

280 

8.6401 

281 

1.3599 

9- 

9996 

3o 

4o 

8.6677 

263 
248 

8.6682 

i.33i8 

9- 

9993 

20 

5o 

8.6940 

8.6945 

i.3o55 

9- 

9993 

0 

10 

3    0 

8.7188 

8 . 7 1 94 

1.2806 

9- 

9994 

0    87 

10 

8.7423 

8.7429 

1 .2571 

9- 

9993 

5o 

20 

8.7645 

212 

8.7652 

223 
213 

1.2348 

9- 

9993 

4o 

3o 

8.7857 

8.7865 

1.2135 

9- 

9992 

3o 

4o 

8.8059 

8.8067 

1.1933 

9- 

9991 

20 

5o 

8.8251 

192 
i8s 

8.8261 

194 
i8s 

I. 1739 

9- 

9990 

10 

4    0 
10 

8.8436 
8.86i3 

177 

8.8446 
8.8624 

.78 

1.1554 
1.1376 

9- 
9- 

99^-9 
99^9 

0 

0    86 

5o 

20 

8.8783 

170 

163 

8.8795 

171 
i6s 

l.I2o5 

9- 

99S8 

4o 

3o 

8.8946 

158 

8.8960 

158 

I .io4o 

9- 

9987 

3o 

4o 

8.9104 

8.9118 

1.0882 

9- 

9986 

20 

5o 

8.9256 

152 

8.9272 

'54 
148 

1.0728 

9- 

9985 

10 

5    0 

8.9403 

8.9420 

i.o58o 

9- 

9983 

0    85 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.Tang. 

L. 

Sin. 

d. 

'     0 

PP 

.  I 

348 

300 

363 

.1 

a35 

313 

185 

.1 

171 

158 

147 

34-8 

30 

26.3 

23-5 

21.3 

18.5 

17.1 

'5- 

3        14.7 

.2 

69.6 

60 

52.6 

.3 

47.0 

42.  t 

37.0 

.2 

34  2 

3«. 

3        29.4 

•3 

104.4 

90 

78.9 

•3 

70-5 

63.5 

55-5 

•3 

51-3 

47- 

*        44- 1 

•  4 

139.2 

I20 

IOS.2 

•4 

94.0 

85.2 

74.0 

•4 

68.4 

63. 

2            58.8 

•5 

174.0 

150 

I3IS 
157.8 

•s 

117."! 

106.  t 

92.5 

•  s 

85.5 

7<)- 

3        73-5 

.6 

208.8 

180 

.6 

141. c 

I27.t 

III.O 

.6 

I02.6 

94. 

3        88.2 

■7 

243.6 

210 

184. 1 

•7 

164.  s 

149.  J 

129.5 

•7 

1 19. 7 

110. 

S      102.9 

.8 

278.4 

240 

210.4 

.8 

188.0 

170.4 

148.0 

.8 

.36.8 

126. 

4      117.6 

.9       3132  1     270    1    236.7     1 

2II.S 

191. 7 

166.5 

142.2  1    132.3    1 

i4o 


FOUR-PLACE   LOGARITHMIC   FUNCTIONS. 

o       ' 

L.  Sin. 

d. 

1 

L.Tang.    d.    L.  Cotg. 

L.  Cos. 

d. 

5     o 

8.9403 

8.9420 

i.o58o 

9.9983 

0    85 

10 

8.9545 

8.9563 

138 

1.0437 

9.9982 

I 

5o 

20 

8.9682 

137 

8.9701 

1 .0299 

9- 

9981 

I 

4o 

>34 

135 

3o 

8.9816 

129 

8.9836 

130 

1 .0164 

9- 

9980 

3o 

4o 

8.9945 

8.9966 

I  .oo34 

9- 

9979 

20 

5o 

9.0070 

125 

9.0093 

0.9907 

9- 

9977 

2 

10 

6    0 

9.0192 

119 

9.0216 

0.9784 

9- 

9976 

I 

0    84 

10 

9.o3i I 

g.o336 

0.9664 

9- 

9975 

5o 

20 

g.0426 

IIS 
113 

9,0453 

"7 
114 

0.9547 

9- 

9973 

2 

4o 

3o 

9.0539 

109 

9.0567 

0.9433 

9- 

9972 

3o 

4o 

9.0648 

9.0678 

108 

0.9322 

9- 

9971 

20 

5o 

9.0755 

107 

9.0786 

0.9214 

9- 

9969 

2 

10 

7    0 

9.0859 

9.0891 

0.9109 

9- 

9968 

0    83 

10 

9 . 096 1 

9.0995 

0.9005 

9- 

9966 

5o 

20 

9. 1060 

99 

9. 1096 

98 
97 

0.8904 

9- 

9964 

2 

4o 

3o 

9.1157 

97 
95 

9.1194 

0.8806 

9- 

9963 

I 

3o 

4o 

9. 1252 

9. 1291 

0.8709 

9- 

9961 

20 

5o 

9.1345 

93 
91 

9.1385 

94 
93 

o.86i5 

9- 

9959 

2 

10 

8    0 

9. i436 

89 

9.1478 

91 

0.8522 

9- 

9958 

0   82 

10 

9.  i525 

9. 1569 

0.843I 

9- 

9956 

5o 

20 

9. 1612 

87 
8S 
84 
82 
80 

9.1658 

89 

87 
86 

0.8342 

9- 

9954 

2 

4o 

3o 

9.1697 

9.1745 

0.8255 

9- 

9952 

2 

3o 

4o 

9.1781 

9.i83i 

0.8169 

9- 

9950 

20 

5o 

9.r863 

9. igiD 

84 
82 
81 

o.8o85 

9- 

9948 

2 

10 

9    0 

9.1943 

9.1997 

o.8oo3 

9- 

9946 

2 

0   81 

10 

9.2022 

9.2078 

80 

0.7922 

9- 

9944 

5o 

20 

9.2100 

78 

9.2i58 

0.7842 

9- 

9942 

2 

4o 

3o 

9.2176 

76 

9.2236 

78 

0.7764 

9- 

9940 

2 

3o 

4o 

9.225t 

75 

9.23i3 

0.7687 

9- 

9938 

20 

5o 

g.2324 

73 
73 

9.2389 

76 
74 

0.7611 

9- 

9936 

2 

2 

10 

10    0 

9.2397 

9.2463 

0.7537 

9- 

9934 

0    80 

L.  Cos.     d. 

1 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

'      0 

PP 

.1 

13 

8 

"5 

117 

.1 

104 

97 

89 

.1 

84 

78 

73 

13 

,8 

12.5 

II.7 

10.4 

9-7 

8.9 

8.4 

7.8 

7-3 

.2 

27 

.6 

25.0 

23.4 

.2 

20.8 

19.4 

,7.8 

.2 

16.8 

.5.6 

.4.6 

•3 

4> 

■4 

375 

35- » 

•3 

31-2 

29.1 

26.7 

•3 

35-2 

234 

21.9 

•4 

55 

.2 

So.o 

46.8 

•4 

41.6 

38.8 

35.6 

•4 

33-6 

31.2 

20.2 

•  s 

bq 

.0 

62.5 

S8.S 

•5    520 

48.5 

44-5 

•s 

42.0 

390 

36.5 

.6 

82 

.8 

750 

70.2 

.6 

t.2.4 

58.2 

53-4 

.6 

5°-4 

46.8 

43-8 

.7 

96 

.6 

87.5 

81.9 

•7 

72.8 

67.9 

62.3 

•7 

58.8 

S4.6 

5I-I 

.8 

lie 

■4 

lOO.O 

03.6 

.8 

83.2 

77.6 

71.2 

.8 

67.2 

62.4 

58.4 

•9     124 

.2     1 1 2. 5  I    IV15.3    1 

.9  1   93.6  I   87.3  1    80.1.     1 

—& 

-^^ 

70.2 

_      65.7 

i4i 


FOUR.PLACE    LOGARITHMIC    FUNCTIONS. 


O        ' 

L.  Sin. 

d. 

L.Tangr.j  d. 

L.  Cotg. 

L.  Cos.  j  d. 

10    o 

lO 

20 

9.2397 
9.2468 
9.2538 

71 
70 

9.2463 
9.2536 
9.2609 

73 
73 

0.7537 
0.7464 
0.7391 

9- 
9- 
9- 

9934 
9931 
9929 

3 

2 

0    80 

5o 

4o 

3o 
4o 
5o 

9.2606 
9.2674 
9.2740 

68 
66 
66 
64 
64 
63 
61 
61 
60 
59 
58 

9.2680 
9.2750 
9.2819 

70 

69 

68 
66 
67 

65 
64 
63 
63 
61 
61 
61 

0.7320 
0.7250 
0.7181 

9- 
9- 
9- 

9927 
9924 
9922 

2 
3 

2 

3o 
20 
10 

11     0 

10 
20 

9.2806 
9.2870 
9.2934 

9.2887 
9.2953 
9.3020 

0.7113 
0.7047 
0.6980 

9- 
9- 
9- 

9919 
9917 
9914 

3 

2 

3 

0    79 

5o 
4o 

3o 
4o 
5o 

9.2997 
9.3o58 
9.3119 

9,3o85 
9.3149 
9.3212 

0.6915 
o.685i 
0.6788 

9- 
9- 
9- 

9912 
9909 
9907 

3 

3 
2 

3 
3 

2 

3o 
20 
10 

12    0 

10 
20 

9.3179 
9.3238 
9.3296 

9.3275 
9.3336 
9.3397 

0.6725 
0.6664 
o.66o3 

9- 
9- 
9- 

9904 
9901 
9899 

0    78 

5o 
4o 

3o 
4o 
5o 

9.3353 
9.3410 
9.3466 

57 
56 
55 
54 
54 

9.3458 
9.3517 
9.3576 

59 
59 
58 
57 
57 
56 
55 
55 
54 
53 
53 

0.6542 
0.6483 
0.6424 

9- 
9- 
9- 

9896 
9893 
9890 

3 
3 

3 
3 
3 
3 

3o 

20 
10 

13    0 

10 
20 

9.3521 
9.3575 
9.3629 

9.3634 
9.3691 
9.3748 

0.6366 
o.63o9 
0.6252 

9- 

9 

9- 

9887 
9884 
9881 

0    77 

5o 
4o 

3o 

4o, 

5o 

9.3682 

,9.3734 

9.3786 

52 
52 
51 
50 
SO 

9.3804 
9.3359 
9.3914 

0.6196 
o.6i4i 
0.6086 

9- 

9 

9 

9878 
9875 
9872 

3 
3 
3 
3 
3 
3 

3o 
20 
10 

14    ^ 

10 
20 

9.3837 
9.3887 
9.3937 

9.3968 
9.4021 
9.4074 

o.6o32 
0.5979 
0.5926 

9 
9 
9 

9869 
9866 
9863 

0    76 

5o 
4o 

3o 
4o 
5o 

9.3986 
9.4035 
9.4083 

49 
49 
48 

47 

9.4127 
9.4178 
9.4230 

53 
51 
52 
51 

0.5873 
0.5822 
0.5770 

9 
9 
9 

9859 

9856 
9853 

4 
3 
3 
4 

3o 
20 
10 

15    0 

9.4i3o 

9.4281 

0.5719 

9-9849 

0    75 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

'      0 

PP 

.1 

.2 
•3 

71 

68 

66 

.1 

.2 
•3 

64 

61 

58 

.1 

.2 

•3 

55 

S3 

51 

71 
14.2 
21.3 

6.8 
13-6 
20.4 

6.6 
13.2 
19.8 

6.4 
12.8 
19.2 

6.1 
12.2 
18.3 

5-8 
11.6 
17.4 

5-5 
II. 0 
16.5 

5-: 
10.  ( 
15c 

f         5« 

>  10.3 

>  153 

•4 

•5 
.6 

28.4 

35-5 
42.6 

27.2 
340 
40.8 

26.4 
33.0 
39-6 

•4 
•5 

.6 

25.6 
32.0 
38.4 

24.4 
30-5 
36.6 

23.2 
29.0 
34-8 

•4 

•5 

33.0 
27s 
330 

21.' 

36. 

3i.i 

J       20.4 
1       3ao 

:I 

49-7 

56.8 
637 

47.6 
544 
61.2 

46.2 
52.8 

^ 

44.8 

42.7 
48.8 

40.6 
46.4 
52.2 

•7 

.i 

38.S 
44.0 

37- 
42. 

4^ 

I       40.8 

142 


FOUR-PLACE    LOGARITHMIC    FUNCTIONS 

O          ' 

L.  Sin.      d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

15    o 

9.4i3o 

47 

9.4281 

0.5719 

9.9849 

0    75 

lO 

9.4177 

9.4331 

0,5669 

9 

9846 

5o 

20 

9.4223 

46 
46 
45 

9.4381 

0.5619 

9 

9843 

3 

40 

3o 

g.4269 

9.4430 

49 
48 

48 

0.5570 

9 

9839 

4 

3o 

4o 

9.4314 

9.4479 

0.552I 

9 

9836 

20 

5o 

9.4359 

45 
44 

9.4527 

0.5473 

9 

9832 

4 
4 

10 

16    0 

9.4403 

9.4575 

0.5425 

9- 

9828 

0    74 

10 

9.4447 

9.4622 

0.5378 

9 

9825 

5o 

20 

9.4491 

44 
42 

9.4669 

47 
47 

o.533i 

9 

9821 

4 
4 

4o 

3o 

9.4533 

9.4716 

46 

0.5284 

9 

9817 

3o 

4o 

9.4576 

9.4762 

0.5238 

9 

9814 

20 

5o 

9.4618 

42 
41 

9.4808 

46 

45 

0.5192 

9 

9810 

4 

4 

10 

17    0 

9.4659 

9.4853 

o.5i47 

9- 

9806 

4 

0    73 

10 

9.4700 

9.4898 

0.5 1 02 

9 

9802 

5o 

20 

9.4741 

40 

9.4943 

45 
44 

o.5o57 

9- 

9798 

4 

4 

40 

3o 

9.4781 

9.4987 

o.5oi3 

9 

9794 

3o 

4o 

9.4«2i 

9.5o3i 

0.4969 

9- 

9790 

20 

DO 

9.4861 

40 

9.5075 

0.4925 

9- 

9786 

4 

10 

18    0 

9.4900 

39 
39 

9.5ii8 

43 

0.4882 

9- 

9782 

4 

0    72 

10 

9-4939 

9.5161 

0.4839 

9- 

9778 

5o 

20 

9-4977 

38 
38 

9.5203 

42 

0.4797 

9- 

9774 

4 

4o 

3o 

9.5oi5 

9.5245 

0.4755 

9- 

9770 

3o 

4o 

9.5o52 

9.5287 

0.4713 

9- 

9765 

20 

5o 

9.5090 

38 
36 

9.5329 

42 
41 

0.4671 

9- 

9761 

4 
4 

10 

19    0 

9.5126 

9.5370 

o.463o 

9- 

9757 

0    71 

10 

9.5i63 

37 

9.541 1 

0.4589 

9- 

9752 

5o 

20 

9.5199 

36 

9.5451 

40 

0.4549 

9- 

9748 

4 

4o 

3o 

9.5235 

36 

9.5491 

0.4509 

9- 

9743 

3o 

4o 

9.5270 

9.5531 

0.4469 

9- 

9739 

20 

5o 

9.5306 

36 
35 

9.5571 

40 
40 

0.4429 

9- 

9734 

5 
4 

10 

20    0 

9.5341 

9.5611 

0.4389 

9- 

9730 

0    70 

L.  Cos.     d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

'     0 

PP 

.1 

49 

47 

45 

.1 

44 

43 

4» 

.1 

40 

38 

36 

4-9 

4-7 

4-5 

4-4 

4-3 

4.1 

4.0 

3.8 

3-6 

.2 

9.8 

9-4 

9.0 

.2 

8.8 

8.6 

8,2 

.2 

8.0 

7.6 

7.2 

■3 

'4-7 

14. 1 

13-5 

-3 

13.2 

12.9 

12.3 

•3 

12.0 

11.4 

10.3 

•4 

19.6 

18.8 

18.0 

•4 

.7.6 

17.2 

.6.4 

•4 

16.0 

15.2 

14.4 

■5 

24.5 

23-5 

22.5 

•5 

22. 0 

21.5 

20.5 

•5 

20.0 

19.0 

18.0 

.6 

29.4 

28.2 

27.0 

.6 

26.4 

25.8 

24.6 

.6 

24.0 

22.8 

21.6 

.7 

34-3 

32-9 

31S 

•7 

30.8 

30.1 

28.7 

.7 

28.0 

26.6 

25.2 

.8 

39-2 

37.0 

36.0 

.8 

35-2 

34-4 

328 

.8 

32.0 

30-4 

28.8 

44.1       42.3   1    40.5      1 

•9       39-6  1    38.7 

369 

•9 

36.0  '' 

34-2 

32.4 

i43 


POUR-PLACE    LOGARITHMIC    FUNCTIONS. 

O          ' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

1 

L.Cos.    d. 

20    o 

9.5341 

34 

9.561 1 

39 

0-4389 

9.9730 

5 

0   70 

lO 

9.5375 

9.565o 

o-435o 

9 

9725 

5o 

20 

9.5409 

34 

9.5689 

38 

o.43i I 

9 

9721 

4 

5 

4o 

3o 

9.5443 

9.5727 

39 

0.4273 

9- 

9716 

5 

3o 

4o 

9.5477 

9.5766 

38 
38 

o.4a34 

9- 

97II 

20 

5o 

9.5510 

33 

9.58o4 

0.4196 

9- 

9706 

5 

10 

21    o 

9.5543 

33 

9.5842 

o.4i58 

9- 

9702 

0    69 

lO 

9.5576 

9.5879 

38 

0.4121 

9- 

9697 

5o 

20 

9.5609 

33 

9.5917 

o.4o83 

9- 

9692 

5 

4o 

3o 

9.5fi4i 

32 

9.5954 

37 

o.4o46 

9- 

9687 

5 

So 

4o 

9.5673 

9.5991 

0.4009 

9- 

9682 

20 

5o 

9.5704 

31 

9.6028 

36 
36 

0.3972 

9- 

9677 

5 

10 

22    o 

9.5736 

32 
31 

9.6064 

0,3936 

9- 

9672 

0    68 

lO 

9.5767 

9 . 6 1 00 

36 

0.3900 

9- 

9667 

6 

5o 

20 

9.5798 

31 

9.6i36 

0.3864 

9- 

9661 

4o 

30 

36 

5 

3o 

9.5828 

31 

g.6172 

36 

0-3828 

9- 

9656 

5 

So 

4o 

9.5859 

9.6208 

0.3792 

9- 

965i 

20 

5o 

9.5889 

30 

9.6243 

35 
36 

35 

0.3757 

9- 

9646 

5 

10 

23    o 

9.5919 

30 

9.6279 

0.3721 

9- 

9640 

0    67 

lO 

9.5948 

9.63i4 

0.3686 

9- 

9635 

5o 

20 

9.5978 

30 

9.6348 

34 

0.3652 

9- 

9629 

4o 

So 

9.6007 

29 

9.6383 

34 

0.3617 

9- 

9624 

So 

4o 

9.6o36 

9.6417 

0.3583 

9- 

9618 

20 

5o 

9.6065 

29 

9.6452 

35 

0-3548 

9- 

961S 

5 

10 

24   o 

9.6093 

28 

9.6486 

34 
34 

o-35i4 

9- 

9607 

0    66 

lO 

9.6121 

9.6520 

o-348o 

9- 

9602 

5o 

20 

9.6149 

28 
28 

9-6553 

33 
34 

0-3447 

9- 

9596 

6 

4o 

So 

9.6177 

9.6587 

o.S4iS 

9- 

9590 

6 

So 

4o 

9.6205 

9.6620 

o.338o 

9- 

9584 

20 

5o 

9.6232 

27 
27 

9-6654 

34 
33 

0-3346 

9- 

9579 

5 
6 

10 

25    o 

9.6259 

9.6687 

o-33i3 

9- 

9573 

0    65 

L.  Cos. 

d. 

L.Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

'     0 

PP 

.1 

39 

3-9 

37 

3-7 

35 

.1 

34 

3-4 

33 

3-3 

39 

.1 

31 

30 

39 

3-5 

3-2 

31 

30 

P 

.2 

7.8 

7-4 

7.0 

.2 

6.8 

6.6 

6.4 

.2 

6.2 

6.0 

•3 

11.7 

II. I 

10-5 

•3 

10.2 

9-9 

9.6 

•3 

9-3 

9.0 

8-7 

•4 

1.S.6 

14.8 

14.0 

•4 

13.6 

13-2 

12.8 

•4 

12.4 

12.0 

11.6 

•5 

»9-5 

i»S 

»7-5 

•5 

17.0 

16.S 

16.0 

•5 

15-5 

150 

J4  5 

.6 

23.4 

22.2 

21.0 

.6 

20.4 

19.8 

19.2 

.6 

18.6 

18.0 

17-4 

•  7 

27.3 

25-9 

24.5 

•7 

23.8 

23.1 

22.4 

■7 

21.7 

21.0 

20.3 

.8 

3I.1 

29.6 

28.0 

.8 

27.2 

26.4 

95-6 

.8 

24.8 

24.0 

23.2 

28.8 

.9  1    27.9  1    27.0 

26. 1 

1 44 


FOUR-PLACE    LOGARITHMIC    FUNCTIONS. 

o         ' 

L.  Sin. 

d. 

L.Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

25   o 

9.6259 

27 

9.6687 

33 

o,33i3 

9.9573 

6 

0   65 

lo 

9.6286 

9.6720 

0.3280 

9- 

9567 

5o 

20 

9.63i3 

27 

9.6752 

32 
33 

0.3248 

9- 

9561 

6 

4o 

3o 

9.6340 

26 

9.6785 

o.32i5 

9- 

9555 

6 

3o 

4o 

9.6366 

9.6817 

o.3i83 

9- 

9549 

20 

5o 

9.6392 

26 
26 

9.685o 

33 
32 

o.3i5o 

9- 

9543 

6 

10 

26    0 

9.6418 

9.6882 

o.3ii8 

9- 

9537 

0    64 

10 

9.6444 

26 
25 
26 

9.6914 

o.3o86 

9- 

9530 

5o 

20 

9.6470 

9.6946 

32 

o.3o54 

9- 

9524 

^ 

4o 

3o 

9.6495 

9.6977 

32 

o.3o23 

9- 

9518 

6 

3o 

4o 

9.6521 

9.7009 

0.2991 

9- 

9512 

20 

5o 

9.6546 

24 

9.7040 

31 
32 

0.2960 

9- 

95o5 

7 
6 

10 

27    0 

9.6570 

25' 

9.7072 

0.2928 

9- 

9499 

7 
6 

0   63 

10 

9.6595 

9.7103 

0.2897 

9- 

9492 

5o 

20 

9.6620 

25 

9.7134 

31 

0.2866 

9- 

9486 

4o 

3o 

9.6644 

9.7165 

0.2835 

9- 

9479 

6 

3o 

4o 

9.6668 

9.7196 

0.2804 

9- 

9473 

20 

5o 

9.6692 

24 
24 

9.7226 

30 
31 

0.2774 

9- 

9466 

7 
7 

10 

28    0 

9.6716 

9.7257 

0.2743 

9- 

9459 

6 

0   62 

10 

9.6740 

9.7287 

0.2713 

9- 

9453 

5o 

20 

9.6763 

23 

9.7317 

30 

0.2683 

9- 

9446 

7 

4o 

So 

9.6787 

9.7348 

31 

0.2652 

9- 

9439 

7 

3o 

4o 

9.6810 

9.7378 

0.2622 

9- 

9432 

20 

5o 

9.6833 

23 

23 

9.7408 

30 
30 

0.2592 

9- 

9425 

7 

7 

ID 

29    0 

9.6856 

9.7438 

0.2562 

9- 

9418 

0    61 

10 

9.6878 

9.7467 

29 

0.2533 

9- 

941 1 

5o 

20 

9.6901 

23 

9.7497 

30 

o.25o3 

9- 

9404 

7 

4o 

3o 

9.6923 

9.7526 

29 

0.2474" 

9- 

9397 

7 

3o 

4o 

9.6946 

9.7556 

0.2444 

9- 

9390 

20 

5o 

9.6968 

22 
22 

9.7585 

29 
29 

0 . 24 1 5 

9- 

9383 

7 
8 

10 

30    0 

9.6990 

9.7614 

0.2386 

9- 

9375 

0   60 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.Tang. 

L.  Sin. 

d. 

'     0 

PP 

.1 

38 

27 

26 

.1 

as 

24 

»3 

.1 

22 

7 

6 

2.8 

2-7 

2.6 

2-5 

2.4 

2.3 

2.2 

0.7 

0.6 

.2 

V6 

5-4 

5.2 

.2 

50 

4.8 

4.6 

.2 

4.4 

1.4 

1.2 

•3 

8.4 

8.1 

7.8 

•3 

7-5 

7-2 

0.9 

•3 

6.6 

2.1 

1.8 

•  4 

II. 2 

10.8 

10.4 

•4 

10. 0 

q.6 

9.2 

•4 

8.8 

2.8 

2.4 

•  5 

14.0 

13  5 

13.0 

•5 

12.  S 

12.0 

ii-S 

•5 

II.O 

3-5 

^■% 

.6 

16.8 

16.2 

15.6 

.6 

iS-o 

14.4 

13.8 

.b 

13.2 

4.2 

3-6 

.7 

19.6 

18.9 

18.2 

•7 

i7-"i 

16.8 

"    16. 1 

■7 

iS-4 

4.9 

*o 

.8 

22.4 

21.6 

20.8 

.8 

20.0 

19.2 

18.4 

.8 

17.6 

I" 

4.8 

9 

25.2  <    24.3 

23- 4 

■9 

22.5 

21.6 

20.7 

.9    1    19.8 

6.3    1      5-4     1 

145 


rOUR-PLACB   LOQARITHMIC    FUNCTIONS. 


o       f 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30   o 

9.6990 

9.7614 

30 

0.2386 

9.9375 

0    60 

lO 

9.7012 

9.7644 

0.2356 

9.9368 

5o 

20 

9.7033 

33 

9.7673 

29 

28 

0.2327 

9.9361 

7 

8 

4o 

3o 

9.7055 

31 

9.7701 

29 

0.2299 

9.9353 

3o 

4o 

9.7076 

9.7730 

0.2270 

9.9346 

20 

5o 

9.7097 

31 

9.7759 

29 
29 

0.2241 

9.9338 

7 

10 

31    0 

9.7118 

9.7788 

28 

0.2212 

9.9331 

g 

0   69 

10 

9.7139 

9.7816 

0.2184 

9.9323 

5o 

20 

9.7160 

21 

9.7845 

29 
38 

o.2i55 

9.9315 

7 

4o 

3o 

9.7181 

9.7873 

29 

0.2127 

9.9308 

g 

3o 

4o 

9.7201 

9.7902 

0.2098 

9.9300 

8 

20 

5o 

9.7222 

9.7930 

28 
28 

0.2070 

9.9292 

10 

32    0 

9.7242 

9.7958 

0.2042 

9.9284 

8 

0    58 

10 

9.7262 

9.7986 

28 

0.2014 

9.9276 

5o 

20 

9.7282 

9.8014 

0. 1986 

9.9268 

4o 

20 

28 

« 

3o 

9.7302 

9.8042 

28 

0.1958 

9.9260 

g 

3o 

40 

9.7322 

9.8070 

0.1930 

9.9252 

20 

5o 

9.7342 

9.8097 

27 
28 

0. 1903 

9.9244 

8 

10 

33    0 

9.7361 

9.8125 

28 

0.1875 

9.9236 

g 

0   57 

10 

9.7380 

9.8153 

0.1847 

9.9228 

5o 

20 

9.7400 

20 
19 

9.8180 

27 
28 

0.1820 

9.9219 

9 
8 

4o 

3o 

9.7419 

9.8208 

0.1792 

9.921 1 

g 

3o 

4o 

9.7438 

9.8235 

0. 1765 

9.9203 

20 

5o 

9.7457 

19 

19 
18 

9.8263 

28 
27 

0. 1787 

9-9'94 

9 
8 

10 

■34    0 

9.7476 

9.8290 

0. 1710 

9.9186 

0    56 

10 

9.7494 

9.8317 

27 

O.I683 

9.9177 

9 

5o 

20 

9.7513 

19 

9.8344 

27 

0. i656 

9.9169 

4o 

3o 

9.7531 

18 

9.8371 

27 

0. 1629 

9.9160 

9 

3o 

4o 

9.7550 

'9 

9.8398 

27 

0. 1602 

9.9151 

9 

20 

5o 

9.7568 

18 
18 

9.8425 

37 
37 

0. 1575 

9.9142 

9 
8 

10 

35    0 

9.7586 

9.8452 

o.i548 

9.9134 

0   55 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin. 

d. 

'     0 

PP 

.1 

3C 

38 

87 

.1 

33 

31 

ao 

.1 

19 

8 

7 

2 

I 

2.8 

2.7 

2.2 

3.1 

2.0 

1.9 

0.8 

0.7 

.3 

,S 

S.6 

5-4 

.3 

4-4 

4.2 

4.0 

.2 

3-8 

1.6 

1.4 

•3 

8 

7 

8.4 

8.1 

•3 

6.6 

0.3 

6.0 

•3 

5-7 

2-4 

2.1 

•4 

11 

6 

II. 2 

10.8 

•4 

8.8 

8.4 

8.0 

•4 

7.6 

3.2 

2.8 

•5 

14 

5 

14.0 

13s 

•5 

11. 0 

10.5 

lO.O 

•5 

9  5 

*•? 

3-5 

.6 

17 

4 

16.8 

l6.2 

.6 

•3-2 

12.6 

12.0 

.6 

11.4 

4.8 

4-3 

•7 

30 

^ 

19.6 

18.9 

•7 

15.4 

14.7 

14.0 

•7 

133 

S.6 

4.9 

.8 

ai 

2 

22.4 

21.6 

.8 

17.6 

16.8 

16.0 

.8 

15.2 

6.4 

56 

9 

26 

I     25.2  1  24.3     1 

^^9 

19.8 

18.9    1      18.0       1 

^^ 

17.1 

7.2    1      6.3     1 

1 46 


FOUR  PLACE    LOGARITHMIC    FUNCTIONS 

O          ' 

L.  Sin. 

d. 

L.Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

35    o 

9.7586 

18 

9.8452 

27 

O.I548 

9.9134 

9 

0    55 

lo 

9 

.7604 

18 

9.8479 

0.  l52I 

9.9125 

5o 

20 

9 

.7622 

18 

9.85o6 

27 

0.1494 

9.91 16 

9 

4o 

3o 

9 

.7640 

9.8533 

26 

0.1467 

9.9107 

3o 

4o 

9 

.7667 

18 

9.8559 

0. 1 44 1 

9.9098 

20 

5o 

9 

.7675 

9.8586 

0. i4i4 

9.9089 

10 

9 

36    0 

9 

.7692 

18 

9.8613 

26 

0.1387 

9.9080 

10 

0    54 

10 

9 

.7710 

9.8639 

0. i36i 

9.9070 

5o 

20 

9 

.7727 

17 
17 

9.8666 

27 
26 

o.i334 

9.9061 

9 

Q 

4o 

3o 

9 

.7744 

9.8692 

26 

o.i3o8 

9.9052 

3o 

4o 

9 

.7761 

9.8718 

0. 1282 

9.9042 

20 

5o 

9 

.7778 

^7 

9.8745 

27 

26 
26 

0. 1255 

9.9033 

9 

10 

37    0 

9 

7795 

16 

9.8771 

0. 1229 

9.9023 

9 

0    53 

10 

9 

.7811 

9.8797 

0. I2o3 

9.9014 

5o 

20 

9 

.7828 

17 

16 

9.8824 

27 
26 

0. 1176 

9 . 9004 

9 

4o 

3o 

9 

7844 

17 
16 
16 
17 

9.885o 

26 

0. ii5o 

9.8995 

3o 

4o 

9 

7861 

9.8876 

26 
26 
26 

0. 1124 

9.8985 

20 

5o 

9 

7877 

9.8902 

0. 1098 

9.8975 

10 

38    0 

9 

7893 

9.8928 

0. 1072 

9.8965 

0    52 

10 

9 

7910 

9.8954 

o.io46 

9.8955 

5o 

20 

9 

7926 

15 

9.8980 

26 
26 

0. 1020 

9.8945 

10 

4o 

3o 

9 

7941 

16 

9.9006 

26 

0.0994 

9.8935 

3o 

4o 

9 

7957 

9.9032 

C.0968 

9.8925 

20 

5o 

9 

7973 

9.9058 

0.0942 

9.8915 

10 

39    0 

9 

7989 

IS 

9.9084 

26 

0.0916 

9.8905 

0    51 

10 

9 

8004 

9.91 10 

0.0890 

9.8895 

5o 

20 

9 

8020 

15 

9.9135 

25 
26 

0.0865 

9.8884 

10 

4o 

So 

9 

8o35 

9.9161 

26 

0.0839 

9.8874 

3o 

4o 

9 

8o5o 

9.9187 

0.081 3 

9.8864 

20 

5o 

9 

8066 

16 

9.9212 

25 

0.0788 

9.8853 

10 

15 

26     ■ 

40    0 

9 

8081 

9.9238 

0.0762 

9.8843 

0    50 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.Tang. 

L.  Sin. 

d. 

'     0 

PP 

.1 

26 

25 

18 

.1 

17   !   16 

15 

.1 

II 

10 

9 

2.6 

2-5 

1.8 

'•7 

1.6 

1-5 

I.I 

I.O 

0.9 

.2 

5-2 

50 

3-6 

.2 

3-4 

3-? 

3.0 

.2 

2.2 

2.0 

1.8 

■3 

7.8 

7-5 

5-4 

•3 

S-« 

4.8 

4-5 

•3 

3-3 

30 

2.7 

•4 

10.4 

10. 0 

7.2 

•4 

6.8 

6.4 

6.0 

•4 

4.4 

4.0 

3-6 

•s 

13.0 

J2.S 

9.0 

■5 

a.'i 

8.0 

7-5 

•s 

55 

.50 

4-5 

.6 

15.6 

ISO 

10.8 

.6 

10.2 

9.6 

9.0 

.6 

6.6 

6.0 

5-4 

•7 

l8.2 

>7-S 

12.6 

•7 

II. 9 

II. 2 

10.5 

•7 

7-7 

7.0 

6.3 

.8 

20.8 

20.0 

14.4 

.8 

13.6 

12.8 

12  0 

.8 

8.8 

8.0 

7-2 

■9        23-4  1    22-5 

16.2 

■9 

15-3   1    '4-4 

'3  5 

.^ 

8,1 

147 


POUR-PLACE   LOGARITHMIC   FUNCTIONS. 


L.  Sin. 


d. 


L.Tang. 


d. 


L.  Cotg. 


L.  Cos. 


d. 


40    o 

lO 

20 

3o 
4o 
5o 


41 


10 
20 

3o 
4o 
5o 


42 


10 
20 

3o 
4o 
5o 


43 


10 
20 

3o 
4o 
5o 


44    o 

10 
20 

3o 
4o 
5o 

46    o 


PP 


9.8081 
9.8096 
9. Bill 

9.8125 
9.8140 
9.8r55 


9.8169 
9.8184 
9.8198 

9.8213 
9.8227 
9.8241 


9.8255 
9.8269 
9.8283 

9.8297 
9.83II 
9.8324 


9.8338 
9.835i 
9.8365 

9.8378 
9.8391 
9.84o5 


9.8418 
9.8431 
9-8444 

9.8457 
9.8469 
9.8482 

9.8495 


L.  Cos. 


36 

2.6 

5  2 

78 

10.4 

130 

15.6 

183 

20.8 


2-5 

so 
7-5 

lO.O 
12  5 

150 

17  5 
20.0 

22-5 


15 

>S 
M 
'S 
IS 

M 
IS 
14 
15 
14 
14 
14 
14 
14 
14 
14 
13 
14 
13 
'4 
13 
13 
14 
•3 
13 
13 
13 

13 
13 
13 

d. 


92  38 
9264 
9289 

93i5 
9341 
9366 


9392 
9417 
9443 

9468 
9494 
9519 


9544 
9570 
9595 

9621 
9646 
9671 


9697 
9722 
9747 

9772 
9798 
9823 


9S48 
9874 
9899 

9924 

9949 
9975 


L.  Cotg. 


15 
30 

4-5 

60 
7  5 
9.0 

10.5 
12.0 
135 


36 

'5 

36 
36 

25 
36 

25 

36 

25 
36 

25 

25 

26 

25 
26 

25 
25 

36 

25 
25 
25 

25 

25 

25 

26 

25 

25 
25 
26 

25 
d. 


5.6 
7.0 
8.4 

9.8 
II. 2 

12.6 

i48 


.0762 
,0736 
.071 1 

,o685 
0659 
o634 


,0608 
,o583 
,0557 

,o532 
,o5o6 

,o48i 


,o456 
.  o43o 
,  o4o5 

,0379 
,o354 
,0329 


o3o3 
.0278 
,0253 

0228 
,0202 

0177 


,Ol52 

0126 

OIOI 

0076 
,oo5i 
,0025 


9.8843 
9.8832 
9.8821 

9.8810 
9.8800 
9.8789 


9.8778 
9.8767 
9.8756 

9.8745 
9.8733 
9.8722 


9.8711 
9.8699 

9.8688 

9.8676 
9.8665 
9.8653 


9.8641 
9.8629 
9.8618 

9.8606 
9.8594 
9.8582 


L.  Tang. 


9.8569 
9.8557 
9.8545 

9.8532 
9.8520 
9.8507 

9.8495 


L.  Sin. 


13 

2.6 


5-2 

6.5 

7-8 

9.1 
104 
"7 


1.2 
2-4 
3-6 


d. 


2.3 

3-3 

4-4 
5-5 
6.6 

7  7 
8.8 

99 


TABLE   VII 


FOUR-PLACE 

NATURAL    TRIGONOMETRIC 

FUNCTIONS 

TO   EVERY  TEN    MINUTES 


FOUR-PLACE 

NATURAL   FUNCTIONS 

O         ' 

Sin. 

d. 

Tang. 

d. 

Cotg.      d. 

Cos. 

d. 

0     o 

, 0.0000 

0 . 0000 

infinit. 

I .0000 

0  90 

lO 

0.0029 

29 

0.0029 

29 

343.7737 

\ . 0000 

5o 

20 

o.oo58 

29 

o.oo58 

29 
29 

171.8854 

I .0000 

0 

0 

4o 

3o 

0.0087 

29 

0.0087 

29 

ti4.5887 

I .0000 

3o 

4o 

o.oi 16 

o.oi 16 

85.9398 

0.9999 

' 

20 

5o 

0.0145 

2y 

0.0145 

29 

68.7501 

1 1 4601 

0.9999 

0 

10 

3° 

30 

1 

1     0 

0.0175 

0.0175 

29 

57.2900 

8 1 861 

0.9998 

0  89 

10 

0.0204 

0.0204 

49. 1039 

0 . 9998 

5o 

20 

0.0233 

29 
29 

0.0233 

29 
29 

42.9641 

6:398 
47756 

0.9997 

0 

4o 

3o 
40 

0.0262 
0.0291 

29 

0.0262 
0.0291 

29 

38.1885 
34.3678 

38207 

0.9997 
0 . 9996 

I 

3o 
20 

5o 

0.0320 

■■"i 

0. o32o 

29 

3i .2416 

31262 

0.9995 

I 

10 

^V 

-ly 

26053 

1 

2    0 

0.0349 

29 

0.0349 

29 

28.6363 

0 . 9994 

0  88 

10 

0.0378 

0.0378 

26.4316 

0 . 9993 

5o 

20 

0.0407 

29 
29 

0.0407 

29 
30 

24.5418 

18898 
16380 

0 . 9992 

2 

4o 

3o 

o.o436 

29 

0.0437 

29 

22.9038 

0 . 9990 

3o 

4o 

o.o465 

o.o466 

21 .4704 

14334 

0 . 9989 

20 

5o 

0.0494 

29 

0.0495 

29 

20.2056 

12648 

0.9988 

I 

10 

3    0 

o.o523 

29 

o.o524 

29 

19.081 1 

0.9986 

0  87 

10 

o.o552 

o.o553 

18.0750 

0.9985 

5o 

20 

o.o58i 

^9 
29 

o.o582 

29 
30 

17.1693 

9057 
8194 

0.9983 

2 
2 

4o 

3o 

0.0610 

30 

0.0612 

29 

1 6 . 3499 

7451 

0.9981 

3o 

4o 

o.o64o 

0.064 1 

i5.6o48 

0.9980 

20 

5o 

0.0669 

29 

0.0670 

29 

14.9244 

6237 

0.9978 

2 

ID 

•^9 

29 

4    0 

0.0698 

29 

0.0699 

30 

14.3007 

5740 

0.9976 

2 

0  86 

10 

0.0727 

0.0729 

13.7267 

5298 

0.9974 

5o 

20 

0.0756 

■iy 

0.0758 

29 

1 3 . 1 969 

0.9971 

3 

4o 

29 

29 

490; 

2 

3o 
4o 

0.0785 
0.08 1 4 

29 

0.0787 
0.0816 

29 

12.7062 

I2.25o5 

4557 

0.9969 
0.9967 

2 

3o 

20 

5o 

0.0843 

■J9 

o.o846 

3" 

II .8262 

4243 

3961 

0 . 9964 

3 

10 

5     0 

0.0872 

0.0875 

1 1 .4301 

0.9962 

0  85 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'     0 

PP 

26053 

16380 

"245 

.1 

S 
i 

194 

623V 

4907 

.1 

3961 

30 

29 

2605 

1638 

1125 

ii9.4 

623.7 

490.7 

396.1 

3-0 

2.9 
58 

.2 

5211 

3276 

2249 

.2 

1638.8 

1247.4 

981-4 

.2 

7g?^ 

6.0 

•  3 

7816 

4914 

3374 

■3 

2458.3 

1871.1 

1472.1 

•3 

1188.3 

9.0 

8.7 

4 

10421 

6552 

4498 

•4 

3277- 6 

2494.8 

1962.8 

•4 

1584.4 

12.0 

11.6 

•  5 

13027 

8190 

5623 

•5 

4097.0 

3"8.5 

2453-5 

•5 

1980. 5 

I5-0 

14.S 

.6 

15632 

9828 

6747 

.6 

4916.4 

3742.2 

2944.2 

.6 

2376.6 

18.0 

"7-4 

•7 

18237 

1 1466 

7872 

•7 

5735-8 

4365-9 

3434-9 

■7 

2772.7 

21.0 

20.3 

.8 

20842 

13104 

8996 

.8 

6555-2  1  4989-6 

3925.6 

.8 

3168.8 

24.0 

23.2 

.9   1  23448   1   14742 

10121 

•9  '  7374  6     56133  '  4416  3 

.9  1  3564.9  1    27.0 

26.1 

i5o 


POUR-PLACE 

NATURAL    FUNCTIONS. 

0         ' 

Sin. 

d. 

Tang, 

d. 

Cotg. 

d. 

Cos. 

d. 

5    o 

0.0872 

29 

0.0875 

29 

11.4801 

3707 

0.9962 

3 

0   85 

lO 

0.0901 

28 

0.0904 

II .0594 

0.9959 

5o 

20 

0.0929 

29 

0.0934 

29 

10.7119 

3475 
3265 

0,9957 

3 

4o 

3o 
4o 

0.0958 
0.0987 

29 

0.0963 
0.0992 

29 

10. 3854 
10.0780 

3074 

0.9954 

o.ggSi 

3 

80 
20 

5o 

0. 1016 

29 

0. 1022 

9.78*^2 

2738 

0.9948 

3 

10 

6    0 

0. io45 

29 

0. io5i 

29 

9.5i44 

0.9945 

0    84 

10 

0. 1074 

0. 1080 

9.2553 

0,9942 

5o 

20 

0. 1  io3 

29 

0. 1 1 10 

3U 
29 

9.0098 

2455 
2329 

0,9989 

3 
3 

4o 

3o 

0. 1182 

29 

0. 1 139 

30 

8.7769 

0,9986 

4 

80 

4o 

0. 1 161 

0. 1 169 

8.5555 

0.9982 

60 

5o 

0. I igo 

^y 

0. I ig8 

■^9 

8.845o 

0,9929 

3 

10 

7    0 

10 
20 

0. 1219 

0. 1248 
0. 1276 

29 
28 
29 

0. 1228 
0. 1257 
0.1287 

29 

30 
^0 

8.1443 
7.9530 
7,7704 

i9«3 
1826 
1746 

0.9925 
0.9922 
0.9918 

3 
4 
4 

0    83 

•5o 

4o 

So 

o.i3o5 

0.1817 

29 

7.5958 

0.9914 

3 

80 

4o 

0.  i334 

0.1 346 

7,4287 

1600 

0,9911 

20 

5o 

O.I363 

■iy 

0.1376 

30 
29 

7.2687 

0.9907 

4 

10 

29 

'533 

4 

8    0 

10 

0. 1392 
0. 1421 

29 
28 

0. i4o5 
0.1435 

30 

7. II 54 
6.9682 

1472 

0.9908 
0.9899 

4 

0    82 

5o 

20 

0.1449 

0. i465 

6.8269 

'413 

0.9894 

5 

4o 

29 

3" 

'357 

4 

3o 

0.1478 

29 

0. 1495 

29 

6.6912 

0.9890 

3o 

4o 

0. i5o7 

0. i524 

6.56o6 

0.9886 

20 

5o 

o.i536 

29 
28 
29 

o.i554 

30 

6.4348 

1258 

0.9881 

5 

10 

9    0 

0. 1 564 

o.i584 

30 

6.3i38 

1 168 

0.9877 

5 

0    81 

10 

0.1593 

0. i6i4 

6. 1970 

1126 

0.9872 

5o 

20 

0. 1622 

29 

0. 1644 

30 

6.0844 

0.9868 

4 

4o 

2« 

29 

1086 

5 

3o 

0.  i65o 

0.1673 

5.9758 

1050 

0.9868 

5 

80 

4o 

0. 1679 

0. 1708 

5.8708 

0.9858 

20 

5o 

0. 1708 

28 

0.1733 

5.7694 

981 

0.9853 

5 

10 

10    0 

0. 1736 

0. 1763 

5.6718 

0.9848 

0   80 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'     0 

PP 

.1 

2738 

1533 

981 

.1 

30 

89 

2 

2 

3 

8 

.1 

1 

5      !     4 

3 

273.8 

153-3 

98.1 

3-0 

2-9 

0.5      0.4 

0.3 

•  2 

547-6 

306.6 

196.2 

.2 

6.0 

5-8 

5 

6 

.2 

I.O 

0.8 

0.6 

•3 

821.4 

459-9 

294.3 

•3 

9.0 

8.7 

8 

4 

-3 

1-5 

1.2 

0.9 

•4 

109S.2 

613.2 

392-4 

•4 

12.0 

11.6 

II 

2 

•4 

2.0 

1.6 

1.2 

•  5 

1369.0 

766.5 

490.5 

•5 

15.0 

14-5 

14 

0 

•5 

2.5 

2.0 

1-5 

.6 

1642.8 

919.8 

588.6 

.6 

18.0 

174 

16 

8 

.6 

3-0 

2-4 

1.8 

•7 

1916.6 

1073. 1 

686.7 

•7 

21.0 

20.3 

•Q 

6 

•7 

3-5 

2.8 

2.1 

.8 

2190.4 

1226.4 

784.8 

.8 

24.0 

23.2 

22 

4 

.8 

4.0 

3-2 

2.4 

.9  '  2464.2  I  137Q.7 

882.g  1      .9  '    27.0    1    26.1     !    25 

2    1      .9  1     4.5     1      3.6 

2-7 

i5i 


FOUR-PLACE 

NATURAL 

FUNCTIONS 

0        ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

10    o 

0.1736 

29 

0.1763 

30 

5.6713 

0.9848 

0  80 

lO 

0. 1765 

0.1793 

5.6764 

0.9843 

5o 

20 

0.1794 

•29 
28 

0.1823 

30 

5.4845 

919 
890 

0.9838 

5 
5 

4o 

3o 
4o 

0. 1822 
o.i85i 

29 

O.I853 
O.I883 

30 

5.3955 
5.3093 

862 
836 

0.9833 
0.9827 

6 

3o 
20 

5o 

0.1880 

29 
28 
29 

0. 1914 

5.2267 

0.9822 

5 

10 

11    o 

0. 1908 

0.1944 

30 

6.1446 

788 

0.9816 

S 

0  79 

lO 

0. 1937 

0.1974 

6.0658 

764 

0.9811 

5o 

20 

0. 1965 

28 

0.2004 

30 

4.9894 

0.9806 

6 

4o 

29 

3' 

742 

6 

3o 

0.1994 

^8 

o.2o35 

3° 

4.9162 

722 

0.9799 

6 

3o 

4o 

0.2022 

o.2o65 

4.8430 

0.9793 

20 

5o 

O.205l 

29 
28 
29 

0.2096 

J" 

4.'/729 

701 

683 
664 

0.9787 

6 

10 

12   o 

0.2079 

0.2126 

30 

4.7046 

0.9781 

6 

0  78 

lO 

0.2108 

0.2166 

4.6382 

0.9776 

5o 

20 

0.2 1 36 

a8 

0.2186 

30 
31 

4.6736 

629 

0.9769 

6 
6 

4o     , 

3o 

0.2164 

29 

0.2217 

4.6107 

613 

0.9763 

6 

3o 

4o 

0.2193 

0.2247 

4.4494 

0.9767 

20 

5o 

0.2221 

0.2278 

4.3897 

597 
582 
568 

0.9760 

7 

10 

13   o 

0.2260 

2y 
28 

0.2309 

30 

4.33i5 

0.9744 

7 

0  77 

lO 

0.2278 

0.2339 

4.2747 

0.9737 

5o 

20 

o.23o6 

28 

0.2370 

31 

4.2193 

540 

0.9730 

7 
6 

4o 

3o 
4o 

0.2334 
0.2363 

29 

0.2401 
0.2432 

3' 

4.1653 
4. 1 126 

527 

0.9724 
0.9717 

7 

3o 
20 

5o 

0.2391 

28 

0.2462 

J" 

4.061 1 

515 

0.9710 

7 

10 

14   o 

0.2419 

28 

0.2493 

31 

4.0108 

491 

0.9703 

V 
7 
7 

0  76 

lO 
20 

0.2447 
0.2476 

29 

0.2624 
0.2655 

31 

3.9617 
3.9136 

481 

0.9696 
0.9689 

5o 
4o 

78 

31 

469 

8 

3o 

o.25o4 

28 

0.2686 

3.8667 

459 

0.9681 

7 

3o 

4o 

0.2.5^2- 

0.2617 

3.8208 

448 

0.9674 

20 

5o 

o.256o 

28 

0.2648 

3' 

3.7760 

0.9667 

7 
8 

d. 

10 

15    o 

0.2588 

0.2679 

3.7321 

0.9669 

0  75 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

'    0 

PP 

.1 

743 

448 

31 

.1 

30 

29 

aS 

.1 

7 

6 

5 

74.2 

44-8 

3-' 

30 

2.9 

2.8 

0.7 

0.6 

0-5 

.2 

148.4 

89.6 

6.2 

.2 

6.0 

^.-5-8 

5-6 

.2 

»-4 

1.2 

1.0 

•3 

222.6 

'34-4 

9-3 

•3 

9.0 

8.7 

8-4 

•3 

2.1 

1.8 

>-5 

•4 

296.8 

179.2 

12.4 

•4 

I 

2.0 

J1.4 

II. 2 

•4 

2.8 

2-4 

2.0 

•5 

37'° 

224.0 

15-5 

•5 

I 

5-0 

14-5 

14.0 

•5 

35 

3-0 

2-5 

.6 

445-2 

368.8 

18.6 

.6 

1 

8.0 

»7-4 

16.8 

.6 

4-2 

3-6 

3.0 

•7 

519-4 

3'3-6 

21.7 

•7 

2 

I.O 

ao.3 

19.6 

•7 

4-9 

4-» 

3-5 

593-6 

3584 

24.8 

.8 

2 

4.0 

232 

22.4 

.8 

5-6 

4.8 

4.0 

■  9      667.8  1   403.2  1    27.9    1     .9  '    a 

7.0        26.1    1    25.2    1 

■9        6.3     1      5.4     1      4.5    1 

162 


FOUR-PLACE 

NATURAL    FUNCTIONS. 

0          ' 

Sin. 

d. 

Tang. 

d. 

Cotg, 

d. 

Cos.      d. 

15    0 

lO 

0.2588 
0.2616 

28 

0.2679 
0.271 1 

32 

3.7321 
3.6891 

430 

0.9659 
0.9652 

7 

0    75 

5o 

20 

0.2644 

28 

0.2742 

3» 
31 

3.6470 

421 
411 

0.9644 

8 

4o 

3o 

0.2672 

28 

0.2773 

32 

3.6059 

4°3 

0.9636 

8 

3o 

4o 

0.2700 

0.2805 

3.5656 

0.9628 

20 

So 

0.2728 

28 
28 

0.2836 

31 

3.5261 

3<35 
387 

0.9621 

7 

10 

3» 

16    0 

0.2756 

28 

0.2867 

32 

3.4874 

379 

0,9613 

8 

0    74 

lO 

0.2784 

28 

0.2899 

3.4495 

0.9605 

So 

20 

0.2812 

0.2931 

32 

3.4124 

37» 

0.9596 

9 

4o 

2« 

3' 

36s 

8 

3o 

0.2840 

28 

0.2962 

3.3759 

357 

0.9688 

8 

3o 

4o 

0.2868 

0.2994 

3.3402 

0.9580 

20 

5o 

0.2896 

28 
28 
28 

0.3026 

32 

3.3o52 

350 

0.9572 

10 

17   o 

0.2924 

o.3o57 

3' 
32 

3.2709 

343 
338 

0.9563 

9 
8 

0    73 

lO 

0.2952 

0. 3089 

3.2371 

0.9SSS 

So 

20 

0.2979 

27 
28 

0.3l2I 

32 
32 

3.2o4i 

33° 
325 

0.9S46 

9 
9 

40 

3o 

0.3007 

28 

o.3i53 

3.1716 

0.9S37 

3o 

4o 

o.3o35 

o.3i85 

32 

3.1397 

319 

0.9528 

9 

20 

5o 

0.3062 

27 

0.3217 

J2 

3.1084 

313 

0.9S20 

10 

18    o 

0.3090 

28 

0.3249 

3'' 

3.0777 

307 
302 

o.gSii 

9 
9 

0    72 

lO 

o.3ii8 

0.3281 

3.0475 

0.9502 

So 

20 

o.3i45 

27 

?8 

o.33i4 

33 
32 

3.0178 

297 
291 

0.9492 

9 

4o 

3o 

0.3173 

0.3346 

2.9887 

287 

0.9483 

9 

3o 

4o 

0.320I 

0.3378 

2.9600 

0.9474 

20 

So 

0.3228 

27 

0.3411 

33 

2.9319 

0.9465 

9 

10 

19   o 

lO 

0.3256 
0.3283 

=7 

0.3443 
0.3476 

33 

2.9042 
2.8770 

272 

0.9455 
0.9446 

9 

0   71 

5o 

20 

o.33ii 

o.35o8 

32 

2.8502 

0.9436 

40 

3o 

0.3338 

27 

0.3541 

33 

2.8239 

263 

0.9426 

10 

3o 

4o 

0.3365 

27 

0.3574 

33 

2.7980 

259 

0.9417 

20 

5o 

0.3393 

28 
27 

0.3607 

33 
33 

2.7725 

255 
250 

0.9407 

10 

10 

20    o 

0.3420 

o.364o 

2.7475 

0.9397 

0    70 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin.      d. 

'     0 

PP 

.1 

«55 

33 

3a 

.1 

31 

38 

27 

I 

10           g 

8 

255 

3-3 

3-2 

31 

2.8 

2.7 

I.O 

0.9 

0.8 

,2 

5>o 

6.6 

6.4 

.2 

6.2 

5-6 

5-4 

2 

2.0 

1.8 

1.6 

•3 

76.5 

9.9 

9.6 

•3 

9-3 

8.4 

8.1 

3 

3.0 

2.7 

2.4 

•4 

I02.0 

132 

12.8 

•4 

I 

2.4 

II. 2 

10.8 

4 

4.0 

3.6 

3-2 

•S" 

"27-5 

>^5 

16.0 

•5 

I 

S-5 

14.0 

13  5 

5 

5-0 

4S 

4.0 

.6 

1530 

19.8 

19.2 

.6 

I 

8.6 

i6.8 

16.2 

6 

6.0 

5-4 

■;8 

•7 

178.5 

23-1 

22.4 

■7 

2 

'•7 

19.6 

18.9 

7 

7.0 

6.3 

5-6 

.8 

204.0       26.4 

25.6 

.8 

2 

4.8 

22.4 

21.6 

8 

8.0 

7.2 

6.4 

•  9       229-5  1    29.7 

28.8           .9  1    2 

7.9    1    25.2    1    24.3    1 

^ 

9         9.0     1      8.1     1      7.2    1 

i53 


FOUR-PLACE 

NATURAL " 

FUNCTIONS 

O          ' 

Sin. 

d. 

Tang.    d. 

Cotg. 

d. 

Cos.      d.l           1 

20    o 

0.3420 

28 

o.364o 

33 

2.7475 

247 

0.9^97 

0   70 

10 

0.3448 

0.3673 

2.7228 

o.9"387 

5o 

20 

0.3475 

27 
27 

0.3706 

33 

33 

2.6985 

243 
239 

0.9377 

10 
10 

4o 

3o 

o.35o2 

27 

0.3739 

2.6746 

235 

0.9367 

3o 

4o 

"0.3529 

28 

0.3772 

2.65i I 

0.9356 

20 

5o 

0.3557 

o.38o5 

33 

2.6279 

0.9346 

10 

10 

21    0 

0.3584 

27 

0.3839 

33 

2.6o5i 

225 

0.9336 

0    69 

10 

o.36i I 

0.3872 

2.5826 

0.9325 

5o 

20 

0.3638 

27 
27 

0.3906 

34 
33 

2.56o5 

219 

0.9315 

10 
II 

4o 

3o 

0.3665 

0.3939 

34 

2.5386 

214 

0.9304 

3o 

4o 
5o 

0.3692 
0.3719 

27 

0.3973 
0.4006 

33 

2.5172 
2.4960 

212 

0.9293 
0.9283 

10 

20 
10 

22    0 

0.3746 

27 

o.4o4o 

34 

2.4751  . 

206 

0^372. 

0    68 

10 

0.3773 

0.4074 

2.4545 

0.9261 

5o 

20 

b. 38oo 

2^ 
27 

0.4108 

34 
34 

2.4342 

io3 
200 

0.925© 

II 
II 

4o 

3o 
4o 

0.3827 
0.3854 

27 

o.4i42 
0.4176 

34 

2.4142 
2.3945 

197 

0.9239 
0.9228 

II 

3o 
20 

5o 

0.388I 

27 
26 

27 

0.4210 

34 

2.3750 

'95 

0.9216 

12 

10 

23    0 

0.3907 

0.4245 

35 
34 

2.3559 

191 

190 

0.9205 

0    67 

10 

0.3934 

0.4279 

2.3369 

0.9194 

5o 

20 

0.3961 

2y 
26 

o.43i4 

3b 
34 

2.3i83 

1 8,5 

0.9182 

II 

4o 

3o 

0.3987 

27 

0.4348 

35 

2.2998 

181 

0.9171 

3o 

4o 

o.4oi4 

0.4383 

2.2817 

180 

0.91 59 

20 

5o 

o.4o4i 

27 
26 

27 
26 

0.4417 

34 

2.2637 

0.9147 

12 

10 

24    0 

10 

0.4067 
0.4094 

0.4452 
0.4487 

35 
35 

2.2460 
2.2286 

•74 

0.9135 
0.9124 

II 

0    66 

5o 

20 

0.4120 

0.4522 

35 

2.21l3 

"73 

0.9112 

4o 

27 

35 

170 

12 

3o 

o.4i47 

26 

0.4557 

35 

2.1943 

168 

0 . 9 1 00 

3o 

4o 

0.4173 

0.4592 

2.1775 

0.9088 

20 

5o 

0.4200 

27 

0.4628 

3O 

2. 1609 

,f,A 

0.9070 

13 

10 

25    0 

o.4a26 

0.4663 

2.1445 

0.9063 

0    65 

Cos. 

d. 

Cotgr. 

d. 

Tang. 

d. 

Sin. 

d. 

'     0 

PP 

.1 

.77 
17.7 

35 

3-5 

34 

3-4 

.1 

33 

3-3 

a? 

36 

.1 

13 

II 

10 

3.7 

2.6 

1.2 

I.I 

I.O 

.2 

35-4 

7.0 

6.8 

.2 

6.6 

5-4 

5-2 

.2 

2-4 

2.2 

2.0 

•3 

53-1 

las 

10.2 

•3 

9.9 

8.1 

7-8 

•3 

3-6 

3-3 

30 

•  4 

70.8 

14.0 

.3.6 

•4 

13.2 

10.8 

"0-4 

•4 

4.8 

4-4 

4.0 

•5 

88.5 

'7-5 

17.0 

•s 

16.  S 

13-5 

13.0 

•5 

6.0 

5-5 

5.0 

.6 

106.  2 

21. 0 

20.4 

.6 

19.8 

16.2 

15.6 

.6 

7.2 

6.6 

6.0 

•7 

123.9 

24-5 

23.8 

•7 

23.1 

18.9 

18.2 

•7 

8.4 

7-7 

7.0 

.8 

141. 6 

28.0 

27.2 

.8 

26.4 

21.0 

208 

.8 

9.6 

8.8 

8.0 

•9 

■59-3 

315 

30.6 

.9       20.7 

243        23.4    1 

.9        10.8     1 

,.9^ 

i54 


FOUR-P] 

[lA 

.CE 

NATURAL    FUNCTIONS. 

0         ' 

Sin. 

d. 

Tang.     d. 

Cotg. 

d. 

Cos. 

d. 

25    o 

0.4226 

27 

0.4663     ^g 

2.1445 

0.9063 

0    65 

10 

0.4253 

0.4699 

2.1283 

160 

.s8 

0.9061 

5o 

20 

0.4279 

26 

0.4734 

35 
36 

2.II23 

0.9038 

'3 
12 

4o 

3o 

o.43o5 

ofi 

0.4770 

36 

2.0966 

156 

0.9026 

13 

3o 

4o 

0.4331 

0.4806 

2.0809 

0.9013 

20 

5o 

0.4358 

27 
26 

26 

0.484I 

3b 

2.o656 

154 

0.9001 

10 

26    o 

0.4384 

0.4877 

3" 
36 

2.o5o3 

0.8988 

0    64 

lO 

0.4410 

0.4913 

^.o353 

0.8976 

5o 

20 

0.4436 

26 

0.4960 

37 
36 

2.0204 

149 
'47 

0.8962 

13 

4o 

3o 

0.4462 

"6 

0.4986 

36 

2.0067 

145 

0.8949 

13 

3o 

4o 

0.4488 

0.6022 

I. 9912 

0.8936 

20 

5o 

o.45i4 

26 

0.6069 

37 

1.9768 

144 

0.8923 

'3 

10 

27    o 

0.4540 

0.6096 

37 

I .9626 

140 

0.8910 

13 

0    63 

lO 

o,4566 

o.6i32 

1.9486 

0.8897 

60 

20 

0.4592 

2b 
25 

0.6169 

37 
37 

1.9347 

139 
137 

0.8884 

'3 
•4 

4o 

3o 
4o 

0.4617 
0.4643 

26 

0.6206 
0.6243 

37 

1 .9210 

1.9074 

136 

0.8870 
0.8867 

13 

3o 
20 

5o 

0.4669 

2b 

0.6280 

37 

I .8940 

134 

0.8843 

14 

10 

28    o 

0.4696 

o.63i7 

37 

1.8807 

0.8829 

0    62 

lO 

0.4720 

^b. 

0.5354 

1.8676 

0.8816 

60 

20 

0,4746 

2b 

0.5392 

3» 
38 

1.8646 

130 
128 

0.8802 

•4 

14 

4o 

3o 

0.4772 

o.543o 

i.84i8 

0.8788 

3o 

4o 

0.4797 

^b 

9.5467 

37 

I .8291 

126 

0.8774 

20 

5o 

0.4823 

2b 

o.55o6 

3*i 

I. 8166 

0.8760 

>4 

10 

29    o 

0.4848 

26 

0.5643 

38 

i.8o4o 

0.8746 

0    61 

lO 

0.4874 

0.558I 

1-7917 

0.8732 

5o 

20 

0.4899 

25 

0.6619 

3*i 

1.7796 

0.8718 

H 

40 

3o 

0.4924 

25 

0.5668 

39 
38 

1.7675 

121 

119 

0.8704 

M 

3o 

4o 

0.4960 

0.6696 

1.7666 

0.8689 

20 

5o 

0.4975 

25 

0.57^5 

39 

1.7437 

119 
116 

0.8676 

'4 

10 

25 

39 

•5 

30    o 

0.  5ooo 

0.6774 

I .7321 

0.8660 

0    60 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'     0 

PP 

.1 

149 

131 

39 

.1 

38 

37 

36 

.1 

SS 

M 

13 

14.9 

13.1 

3-9 

3-8 

3-7 

3-6 

2-5 

1.4 

1-3 

.2 

29.8 

26.2 

7.8 

.2 

7.6 

7-4 

7-2 

.2 

50 

2.8 

2.6 

•3 

44-7 

39-3 

11.7 

■3 

11.4 

II. I 

10.8 

•3 

7-S 

4.2 

3-9 

•4 

59-6 

52-4 

15-6 

•4 

IS-2 

14.8 

14.4 

•4 

lO.O 

5.6 

5.2 

•5 

74-5 

655 

19.5 

•5 

19.0 

18.5 

18.0 

•5 

12.5 

7.0 

6.5 

.6 

89.4 

78.6 

23.4 

.6 

22.8 

22.2 

21.6 

.6 

ISO 

8.4 

7.8 

•7 

104.3 

91.7 

273 

•7 

26.6 

25.9 

25.2 

•7 

I7-S 

9.8 

9.1 

.8 

119.2 

104.8 

31.2 

.8 

30-4 

29.6 

28.8 

.8 

20.0 

II. 2 

10.4 

■9       '34'   '   "79 

35.1    1      .9  1    34.2 

33-3         324 

.9       22.5    1    12.6    1    11.7    1 

i55 


FOUR-PLACE 

NATURAL 

FUNCTIONS 

O          ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

30    o 

o.5ooo 

25 

0.5774 

lS 

I. 7321 

116 

0.8660 

14 

0    60 

lO 

o.5o25 

o.58i2 

1.7205 

0.864a 

5o 

20 

o.5o5o 

2S 

o.585i 

3Q 

I . 7090 

"5 
113 

0.863I 

IS 

15 

4o 

3o 

0.5075 

25 

0.5890 

1.6977 

"3 

0.8616 

3o 

4o 

0.5 1 00 

0.5930 

1.6864 

0.8601 

20 

DO 

o.5i25 

•'b 

0.5969 

3y 

1.6753 

III 

0.8587 

14 

10 

31    0 

o.5i5o 

25 

0.6009 

39 

1.6643 

0.8572 

'5 

0    59 

10 

0.5175 

o.6o48 

1.6534 

0.8557 

5o 

20 

0.5200 

25 

0.6088 

40 
40 

1.6426 

108 
IC7 

0.8542 

15 
16 

4o 

3o 

0.5225 

0.6128 

I .63i9 
1.6212 

0.8526 

3o 

4o 

o.525o 

^b 

0.6168 

4" 

107 

o.85ii- 

'5 

20 

5o 

0:5275 

^b 

0.6208 

40 

I .6107 

'05 

0.8496 

'b 

10 

32    0 

0.5299 

0.6249 

40 

I .6oo3 

103 

0.8480 

15 

0    58 

10 

0.5324 

0.6289 

I .5900 

0.8465 

5o 

20 

0.5348 

2S 

o.633o 

41 
41 

1.5798 

lOZ 

1 01 

o.845o 

'5 
16 

4o 

3o 

0.5373 

0.6371 

4' 

1.5697 

0.8434 

16 

3o 

4o 

0.5398 

o.64i2 

1.5597 

o.84i8 

20 

5o 

0.5422 

M 

0.6453 

4' 

1.5497 

100 

o.84o3 

•5 
16 
16 

10 

33   0 

0.5446 

25 

0.6494 

4' 
42 

I .5399 

98 
98 

0.8387 

0   57 

10 

0.5471 

0.6536 

I .53oi 

0.8371 

16 

5o 

20 

0.5495 

24 

0.6577 

41 

I .52o4 

97 

0.8.355 

4o 

24 

42 

96 

16 

3o 

0.5519 

0.6619 

i.5io8 

0.8339 

16 

3o 

4o 

0.5544 

25 

0.6661 

i.5oi3 

0.8323 

20 

5o 

0.5568 

24 
24 

0.6703 

42 

42 

I. 4919 

94 
93 

o.83o7 

16 
17 

10 

34    0 

0.5592 

24 

0.6745 

42 

1.4826 

93 

0.8290 

16 

0    56 

10 

o.56i6 

0.6787 

1.4733 

0.8274 

16 

5o 

20 

o.564o 

24 

0.6830 

43 

I.464I 

92 

o,8258 

4o 

24 

43 

91 

17 

3o 

0.5664 

0.6873 

i.455o 

0.8241 

16 

3o 

4o 

0.5688 

0.6916 

1.4460 

0.8225 

20 

5o 

0.5712 

24 

0.6959 

43 

1.4370 

90 

0.8208 

'7 
16 

10 

35    0 

0.5736 

0.7002 

I. 4281 

0.8192 

0    55 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin.      d. 

'     0 

.1 

43 

4« 

41 

.1 

40 

as 

24 

.1 

17 

16 

15 

4-3 

4-2 

t' 

1° 

2.S 

'i 

1-7 

1.6 

IS 

.2 

8.6 

8.4 

8.2 

.2 

8.0 

S-o 

4.8 

.2 

3-4 

,3-2 

30 

■3 

12.9 

12.6 

".3 

•3 

I2.0 

7.5 

7.2 

•3 

51 

4.8 

45 

•4 

17.2 

i6.8 

16.4 

•4 

16.0 

lO.O 

9.6 

•4 

6.8  . 

6.4 

6.0 

•5 

2I-5 

21.0 

20.5 

•  5 

20.0 

12.5 

12.0 

•S 

8.5 

8.0 

75 

.6 

25-8 

25-2 

24.6 

.6 

24.0 

15.0 

14.4 

.6 

10.2 

9.6 

9.0 

•7 

30.1 

29.4 

28.7 

•7 

28.0 

17-5 

16.8 

•7 

11.9 

II. 2 

10.5 

.8 

34-4 

33- <> 

32.8 

.8 

32.0 

20.0 

19.2 

.8 

136 

12.8 

12.0 

■9  1    38.7    1    37-8 

369 

.9  1    36.0 

22.5    1    21.6    1 

.^ 

>5  3          14-4 

'3-5 

1 56 


FOUR  PLACE 

NATURAL    FUNCTIONS. 

0         / 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos.      d. 

35    o 

0.5736 

24 

0.7002 

44 

I. 4281 

88 

0.8192 

0    55 

10 

0. 6760 

0.7046 

I. 4193 

87 

0.8175 

5o 

20 

0.5783 

23 

0.7089 

43 

I .4106 

o.8i58 

17 

4o 

^4 

44 

^1 

17 

3o 
4o 

0.5807 
0.583I 

24 

0.7133 
0.7177 

44 

I. 4019 
1.3934 

85 
86 
84 
84 

o.8i4i 
0.8124 

17 

3o 
20 

5o 

0.5854 

23 

0.7221 

44 

1.3848 

0.8107 

J7 

10 

36    o 

0.5878 

23 

0.7265 

45 

1.3764 

0.8090 

17 

0    54 

10 

0.5901 

0.7310 

I.3680 

83 

0.8073 

5o 

20 

0.5925 

^4 

0.7355 

4b 

1.3597 

o.8o56 

'7 

4o 

23 

4S 

83 

'7 

3o 

0.5948 

24 

0.7400 

45 

i.35i4 

82 

0.8039 

18 

3o 

.  4o 

0.5972 

0,7445 

1.3432 

81 
81 

0.8021 

20 

5o 

0.5995 

^3 

0.7490 

4b 
46 

i.335i 

0.8004 

17 

10 

37   o 

0.6018 

23 

0.7536 

I .3270 

80 

0.7986 

17 

0    53 

10 

o.6o4i 

0.7581 

46 

I .3190 

0.7969 

5o 

20 

o.6o65 

0.7627 

i.3iii 

79 

0.7951 

4o 

23 

4t, 

79 

17 

3o 

0.6088 

0.7673 

47 

I .3o32 

78 

0.7934 

18 

3o 

4o 

o.6rii 

0,7720 

1 .2954 

0.7916 

20 

5o 

0.61 34 

23 

0.7766 

40 

1.2876 

78 

0.7898 

10 

38    o 

0.6157 

23 

0,7813 

47 

1.2799 

76 

0.7880 

18 

0    52 

lO 

0.6180 

0.7860 

I .2723 

0.7862 

5o 

20 

0.6202  ' 

22 

23 

0.7907 

47 
47 

I .2647 

76 
75 

0.7844 

18 
18 

4o 

3o 

0.6225 

0.7954 

48 

I .2572 

75 

0.7826 

18 

3o 

4o 

0.6248 

0.8002 

1.2497  ' 

0.7808 

20 

5o 

0.6271 

23 

o.8o5o 

48 

1.2423 

74 

0.7790 

18 

10 

48 

74 

'9 

39    o 

0.6293 

0.8098 

48 

1.2349 

73 

0.7771 

0    51 

lO 

o.63i6 

o,8i46 

I .2276 

0.7753 

5o 

20 

0.6338 

22 

23 

0.8195 

49 
18 

I .2203 

73 
72 

0.7735 

18 
19 

4o 

3o 

0.636I 

0.8243 

I.2l3l 

0.7716 

3o 

4o 

0.6383 

0.8292 

49 

1 ,2059 

72 

0.7698 

20 

5o 

o.64o6 

23 
22 

0.8342 

50 
49 

I. 1988 

71 
70 

0.7679 

19 
'9 

10 

40    o 

0.6428 

0.8391 

I.I9I8 

0.7660 

0    50 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'      0 

PP 

.1 

48 

47 

46 

4.6 

■  I 

45 

44 

23 

.1 

33 

19 

18 

4.8 

4-7 

4-5 

44 

2-3 

2.2 

1.9 

1.8 

.2 

9.6 

9.4 

9^ 

.2 

9.0 

8.8 

4.6 

.2 

4-4 

3.8 

3.6 

•3 

14.4 

14. 1 

13-8 

■3 

13s 

13-2 

6.9 

•3 

6.6 

5-7 

5-4 

•4 

19.2 

•18.8 

18.4 

■4 

18.0 

17.6 

9.2 

•4 

8.8 

7.6 

7.2 

•5 

24.0 

23- 5 

230 

•5 

22.5 

22.0 

"•5 

■5 

II. 0 

9- .5 

9.0 

.6 

.28.8 

28.2 

27.6 

.6 

27.0 

26.4 

13-8 

.6 

13.2 

II. 4 

10.8 

•  7 

33-6 

32-9 

32.2 

•7 

3'.S 

30.8 

16.1 

•7 

15-4 

13-3 

12.6 

.8 

38.4 

37-6 

36.8 

.8 

36.0 

35-2 

.8.4 

.8 

.7.6 

15.2 

14.4 

•9       43-2     1     42-3     1     4I-4    1 

•9 

40.5 

39.6         20.7    1 

.9       19-8    1 

17. 1     1     16.2    1 

i57 


FOUR  PLACE 

NATURAL   FUNCTIONS. 

O          ' 

Sin. 

d. 

Tang. 

d. 

! 

Cotg.      d. 

Cos.      d. 

40    o 

0.6428 

22 

0.8391 

so 

1.1918 

71 
69 

0.7660 

18 

0    50 

lO 

o.645o 

0.844I 

5° 

I. 1847 

0.7642 

5o 

20 

0.6472 

0.8491 

1.1778 

0.7623 

'9 

40 

22 

i" 

70 

iq 

3o 

0.6494 

23 

0.854I 

so 

I. 1708 

68 

0.7604 

'9 

3o 

4o 

o.65i7 

0.8591 

1 . i64o 

0.7686 

20 

5o 

0.6539 

0.8642 

51 

I . 1671 

69 
67 

0.7666 

'9 

10 

41    0 

0.656I 

0.8693 

SI 

1 . i5o4 

68 

0.7547 

19 
'9 

0    49 

10 
20 

0.6583 
0.6604 

21 

0.8744 
0.8796 

52 

1.1436 
1.1369 

67 

0.7628 
0.7609 

60 
4o 

22 

b» 

66 

'9 

3o 

0.6626 

22 

0.8847 

S2 

i.i3o3 

66 

0.7490 

3o 

4o 

0.6648 

0.8899 

1.1237 

0.7470 

20 

5o 

0.6670 

0.8952 

52 

S3 

1 . 1171 

65 
65 

0.7461 

'9 

10 

42    0 

0.6691 

22 

0.9004 

1.1106 

0.7431 

»9 

0   48 

10 

0.6713 

0.9067 

1 .io4i 

0.7412 

60 

20 

0.6734 

22 

0.9110 

S3 

1.0977 

64 
64 

0.7392 

19 

4o 

3o 

0.6766 

21 

0.9163 

S4 

1 . 09 1 3 

63 

0.7373 

3o 

4o 

0.6777 

0.9217 

1 .0860 

0.7353 

20 

5o 

0.6799 

0.9271 

1.0786 

64 
62 
63 

0.7333 

10 

43    0 

0.6820 

0.9326 

ss 

1.0724 

0.7314  1^1 

0   47 

10 

0.684I 

0.9380 

I .0661 

0.7294 

5o 

20 

0.6862 

22 

0.9435 

I .0699 

62 
61 

0.7274 

20 
20 

4o 

3o 

0.6884 

0.9490 

55 

1.0538 

61 

0.7254 

3o 

4o 

0.6905 

0.9646 

56 
56 
S6 

1.0477 

61 
61 
60 

0.7234 

20 

5o 

0.6926 

0.9601 

1 . o4 1 6 

0.7214 

10 

44    0 

0.6947 

0.9667 

I  ..0355 

0.7193 

0   46 

10 

0.6967 

0.9713 

1 .0296 

60 

0.7173 

5o 

20 

0.6988 

0.9770 

57 

I.0235 

0.7163 

4o 

21 

57 

59 

20 

3o 

0.7009 

0.9827 

S7 

1 .0176 

59 

0.7133 

3o 

4o 

0.7030 

0.9884 

58 
58 

I .0117 

0.71 12 

20 

5o 

o.7o5o 

20 

0.9942 

1 .0068 

59 
58 

0.7092 

10 

45    0 

0.7071 

I . 0000 

1 .0000 

0.7071 

0   45 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'     0 

PP 

.1 

57 

55 

54 

.1 

53 

51 

33 

I 

31 

30 

»9 

S-7 

S-5 

5-4 

5-3 

5- 1 

2.2 

2.1 

2.0 

't 

.2 

I 

1.4 

II. 0 

ia8 

.2 

10.6 

10.2 

4-4 

2 

4.2 

4.0 

3-8 

•3 

I 

71 

16. 5 

16.2 

•3 

15.9 

15-3 

6.6 

3 

6.3 

6.0 

5-7 

■4 

2 

2.8 

22.0 

21.6 

•4 

21.3 

20.4 

8.8 

4 

8.4 

8.0 

7.6 

•  5 

2 

«-5 

27- 5 

27.0 

•5 

26.5 

25-5 

II. 0 

5 

10. 5 

lO.O 

9-5 

.6 

3 

♦■2 

330 

324 

.6 

31.8 

30.6 

13.2 

6 

13.6 

12.0 

11.4 

•7 

3 

3Q 

38.5 

37.8 

•7 

37-1 

35-7 

15.4 

7 

14-7 

14.0 

•3-3 

.8 

4 

5.6 

44.0 

43-2 

.8 

424 

40.8 

17.6 

8 

.6.8 

16.0 

•5-2 

•9        5 

'■3        49-5 

48.6 

•9 

47-7 

45.9        19.8    1 

^ 

^ 

18.9        18.0    I    17. 1    1 

i58 


TABLE   VIII. 
SQUARES   AND   SQUARE   ROOTS   OP  NUMBERS. 

SQUARES  OF  INTEGERS  FROM  10  TO  100. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 
20 

3o 

100 
4oo 
900 

121 
44 1 
961 

1 44 

484 

1024 

169 

529 

1089 

196 

576 

ii56 

225 

625 

1225 

256 

676 

1296 

289 

729 

1369 

324 

784 

1 444 

36i 
84 1 

l52I 

4o 
5o 
6o 

1600 
25oo 
36oo 

1681 
2601 
3721 

1764 
2704 
3844 

1849 
2809 
3969 

1936 
2916 
4096 

2025 

3o25 
4225 

21 16 
3i36 
4356 

2209 
3249 
4489 

2  3o4 
3364 
4624 

2401 
3481 
4761 

70 
80 
90 

4900 
64oo 
8100 

5o4i 
656i 
8281 

5i84 
6724 
8464 

5329 
6889 
8649 

5476 
7o56 
8836 

5625 
7225 
9025 

5776 
7396 
9216 

5929 
7569 
9409 

6084 
7744 
9604 

6241 
7921 
9801 

SQUARE  ROOTS  OF  NUMBERS  FROM  0  TO  lO;  AT  INTERVALS  OF  .1, 


N 

.0 

.1 

.2 

.3 

.4 

.6 

.6 

.7 

.8 

.9 

0 

I 
2 
3 

4 
5 
6 

7 

8 

9 

0 

.3i6 

.447 

.548 

.632 

.707 

.775 

.837 

•  .894 

.949 

1. 000 
i.4i4 
1.732 

2.000 

2.236 

2.449 

2.646 
2.828 
3.000 

1/049 
1.449 
1. 761 

2.025 
2.258 

2.470 

2.665 
2.846 
3.017 

1.095 
1.483 
1.789 

2.049 

2.280 
2.490 

2.683 
2.864 
3.033 

i.i4o 
1.517 
1.817 

2.074 

2.3o2 
2.5lO 

2.702 

2.881 

3.o5o 

i.i83 
1.549 
1.844 
2.098 

2.324 

2.53o 

2.720 
2.898 
3.066 

1.225 

i.58i 
1.871 

2. 121 
2.345 
2.55o 

2.739 
2.915 
3.082 

1.265 
1. 612 
1.897 

2.145 
2.366 
2.569 

2.757 
2.933 
3.098 

i.3o4 
1.643 
1.924 

2.168 

2.387 
2.588 

2.775 
2.950 
3.114 

1.342 
1.673 
1.949 

2. 191 

2.4o8 

2.608 
2.793 

2.966 
3.i3o 

1.378 
1.703 
1.975 

2.2l4 
2.429 
2.627 

2.81I 
2.983 

3.146 

SQUARE  ROOTS  OF  INTEGERS  FROM  10  TO  100. 


N 


10 

20 
3o 

4o 
5o 
60 

70 
80 
90 


3.162 

4.472 
5.477 
6.325 
7.071 
7.746 

8.367 
8.944 
9.487 


3.317 
4.583 
5.568 

6.4o3 
7.i4i 
7.810 

8.426 
9.000 
9.539 


2 


3.464 
4.690 
5.657 

6.481 
7.21 1 
7.874 

8.485 
9.055 
9.592 


3.606 

4.796 
5.745 

6.557 
7.280 
7.937 

8.544 
9.1 10 
9-644 


3.742 
4.899 
5.83i 

6.633 
7.348 
8.000 

8.602 
9.165 
9.695 

159 


6 


3.873 
5.000 
5.916 

6.708 
7.416 
8.062 

8.660 
9.220 

9-747 


4.000 
5.099 
6.000 

6.782 
7.483 
8.124 

8.718 
9.274 
9.798 


4.123 
5.196 
6.083 

6.856 
7.55o 
8.185 

8.775 
9.327 
9-849 


8 


4-243 
5.292 
6.164 

6.928 
7.616 
8.246 

8.832 
9.381 
9.899 


4.359 
5.385 
6.245 

7.000 
7.681 
8.307 

8.888 
9.434 
9.950 


TABLE   IX. 

THE   HYPERBOLIC  AND  EXPONENTIAL  FUNCTIONS  OP 
NUMBERS  FROM  0  TO  2.5,  AT  INTERVALS  OF  .1. 


a? 

eoshic 

sinh^K 

tanh  X 

e' 

e-" 

0 

.1 

1 .000 

0 

0 

1 .000 

1 .000 

I  .oo5 

.  100 

.  100 

I .  io5 

.905 

.2 

.3 

1 .020 
1.045 

.201 
.3o5 

.197 
.291 

1 .221 
i.35o 

.819 
.741 

.4 
.5 
.6 

1. 081 
1. 128 
i.i85 

.4ii 

.521 

.637 

.38o 
.462 
.537 

1 .492 
1 .649 
1.822 

.670 
.607 
.549 

•7 

.8 

•9 
1.0 

I .  I 

I  .2 

1.3 

1.255 
1.337 
1.433 

.759 

.888 
1 .027 

.604 
.664 
.716 

2.014 
2.226 
2.460 

.497 
.449 
.407 

1.543 

1. 175 

.762 

2.718 

.368 

1 .669 
1. 811 
1. 971 

1.336 
1 .509 
1.698 

.801 
.834 
.862 

3.oo4 
3.320 
3.669 

.333 
.3oi 
.273 

1.4 
1.5 
1.6 

2.l5l 
2.352 

2.577 

1.904 
2. 129 
2.376 

.885 
.905 
.922 

4.o55 

4.482 
4.953 

.247 

.223 
.202 

1-7 
1.8 
1.9 

2.0 

2.  I 
2.2 
2.3 

2.828 
3.107 

3.418 

2.646 
2.942 
3.268 

.935 

.947 
.956 

5.474 
6.o5o 
6.686 

.i83 
.i65 
.i5o 

3.762 

3.627 

.964 

7.389 

.135 

4.144 

4.568 
5.037 

4.022 
4.457 
4.937 

.970     • 

.976 

.980 

8.166 
9.025 
9.974 

.  122 
.III 
.  100 

2.4 
2.5 

5.557 
6.132 

5.466 
6.o5o 

.984 
.987 

1 1 .023 

12. 182 

.091 
.082 

160 


TABLE    X 

CONSTANTS 

MEASURES    AND    \VEIGHTS 
AND    OTHER   CONSTANTS 


MEASURES    AND    "WEIGHTS 

English  Measures  Metric  Measures 

LENGTH 

lo  millimeterB  (mm.)  =  i  centimeter  (cm.), 
lo  centimeters  =  i  decimeter  (dcm.). 


LENGTH 

XI  inches  (in.)  =  i  foot  (ft.). 

3  feet 

=  I  yard  (yd.). 

\(>%  feet 

:=  I  rod  (rd.l. 

5280  feet 

=  I  mile  (m.). 

6080.3  feet 

=  I  nautical  mile 

S%  yards 

=  I  rod. 

4  rods 

=:i  chain  (ch.). 

I  foot 

=  30.48  centimete 

I  yard 

=     .9144  meter. 

I  mile 

=   1.6093  kilomet 

SURFACB 

144  sq.  inches  =  i  sq.  foot. 
9  sq.  feet  =  i  sq.  yard. 
3oJ^  sq.  yards  =  i  sq.  rod. 


160  sq.  rods 
43560  sq.  feet 
640  acres 

I  sq.  inch 
I  sq.  foot 
I  sq.  yard 
I  acre 


= I  acre. 
=  I  acre. 
=:  I  sq.  mile. 

=  6.4516  sq.  centimeters. 
=  0.0929  sq.  meter. 
=: 0.8361  sq.  meter. 
=  0.4047  hectare. 


VOLUMK 

1728  cu.  inches  =  i  cu.  foot. 

27  cu.  feet  =  I  cu.  yard. 

128  cu.  feet  =  I  cord  (cd). 

I  cu.  inch  =  16.387  cu.  centimeters. 

I  cu.  foot  =  0.028  cu.  meter. 

I  cu.  yard  =  0.7646  cu.  meter. 

I  cord  =  3.625  steres. 


CAPACITY 

1  liq.  gal.  =  3.785    liters  =  231  cu.  in. 
I  dry  gal.  =  4.404    liters  =   268.8  cu.  in. 
I  bushel  =0.3534  bid.  =2150.42  cu.  in. 


AVOIRDUPOIS  WKIGHT 

16  ounces  (oz.)      =  i  pound  (lb.). 
100  lbs.  =  I  hundredweight  (cwt.). 

20  hundredweight  =  i  ton  (T.). 

I  pound  =  .4536  kilo.  =  7000  grains. 
I  ton       =.9071  tonneau  (t). 


TROV   WEIGHT 

I  pound  =  5760  grains  =:  X2  ounces. 


10  decimeters 
10  meters 
10  dekameters 
10  hektometers 


I  meter 


\ 


I  kilometer 


=  I  meter  (m.). 
=  t  dekameter  (dkm.). 
=  I  hektometer  (hkm.). 
=  I  kilometer  (km.). 

1  =  39.37  inches. 
=  3.2808  feet. 
=  0.6214  mile. 


SIJRFACE 

100  sq.  millimeters 
100  sq.  centimeters 

100 


3  sq.  decimeters  | 

100  sq.  meters 
100  ares 

I  sq.  centimeter 

I  sq.  meter        J 

I  are 

I  hektare 


=  I  sq.  centimeter. 
=  i  sq.  decimeter. 
=  I  sq.  meter. 
=  I  centare  (ca.). 
==  I  are  (a.). 
=  1  hektare  (hka.). 

=   0.1550  sq.  inch. 
=   1. 196  sq.  yards. 
=  10.764  sq.  feet. 
=  1076.48  sq.  feet. 
=  2.471  acres. 


VOLUME 

1000  cu.  millimeters  =  i  cu.  centimeter. 
1000  cu.  centimeters  =  i  cu.  decimeter. 
[  =  J  cu.  meter. 


1000  cu.  decimeters ' 


=  I  stere  (st.). 


I  cu.  cantimeter  =  0.061  cu.  inch. 

[  =35.314  cu.  feet. 

1  =   1.308  cu.  yards. 
1  stere  =  0.2759  cord. 


I  cu.  meter 


■    CAPACITY 

100  centiliters  (cl.)  =  i  liter  (1.). 

100  liters  =  I  hektoliter  (hid). 

I  liter  =  1.0567  liq.  qts.  =  i  cu.  dcm. 


METRIC  WEIGHT 

1000  grams  (gm.)  =  i  kilogram  (kilo.). 
1000  kilograms    =  i  tonneau  (t.). 

I  gram  =  J  5-432  grains. 

I  kilogram       =   2.2046  pounds. 
I  tonneau         =   1.1023  tons. 


162 


MEASURES    AND    WHIGHTS—Coniinued 


60  seconds  (")=  •  minute  ('). 
60  minutes  =  i  degree  (°). 
90  degrees       =  i  right  angle. 


—  radians         =  i  right  angle. 


w  =  3.14159263359 
27r  =  6.2831853 
4ir  =12.5663706 

=  1. 0471976 


—  4.1 


3 

4ir 
3 

—  =  0.78539S2 

-^  =  0.5235988 

—  =  0.3183099 

IT 

«•'=    9.8696044 

I 

— -z=.  0.1013212 

Vn  =  1.7724539 
~p^—  0.5641896 


CONSTANTS 


log    TT 

=  0.4971499 

log  27r 

=  0.7981799 

log  4jr 

=  1.0992099 

log  — 

=  0.1961199 

1    "■ 
log  - 

^0.0200286 

logf 

=  0.6220886 

log  — 

=  9.8950899  —  10 

log  I 

=  9.7189986  — 10 

log  — 

=  9.5028501  — 10 

log  w« 

=  0.9942997 

log  1, 

=  9.0057003  —  10 

logVff 

=  0.2485749 

log 

=  9.7514251  —  10 

lOg^TT  =  1.1447299 


e  ^2.718281828459 
M  =0.4342945 

-^  =  2.3025851 


Radian 


r=  57. 295779  5-^ 

I  =  3437-747' 
■  =  206264.8" 


I  degree  =0.0174533  radians 
I  minute  =  0.0002909  radians 
I  second  =  0.0000048  radians 


log^  =0.4342945 

log  M  =  9.6377843  — 10 

log  ^  =  0.3622157 

log  57-2957795  =  1. 7581226 
log  3437-747  =  3-5362739 
log  206264.8  =  5.3144251 


log  0.0174533  =  8.2418774  —  10 
log  0.0002909  =  6.4637261  — 10 
log  0.0000048  =  4.6857749  — 10 


i63 


87-f3 


QA 


Phillips-Elements  of  trigonometry^ 
plane  and  spherical 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


INIVERSITY  of  CATJFORNTA 


f^ 


A     OOP  257  691     6 
5-3  I 


JUL  72 


